Christmas Eve

My memories are organized like a relational database. Holidays are often the key to accessing thoughts of times long ago, although music can also serve as a key phrase to recall specific times and places.

No other holiday is as important for bringing up old times and remembrance than Christmas. My thoughts roll back over data records of Christmas past, of family and friends, of places I’ve been … like the Christmas I spent at a customer location in Philadelphia. While most get vacation time for Christmas, and I’ve been amongst those lucky people most of my life, there was a time that an important IBM customer problem required me to spend the holiday in a strange city. We did break for Christmas and I was home for three days before returning to the humming halls of customer mainframes and data farms. It was a happy visit, we saved the customer's hind quarters, and IBM kept a paying client, largely based on my emergency services.

But that’s not the tale I was thinking about last night. Linda and I always attend the candle-light service at our local church on Christmas Eve. That’s a tradition we’ve had this last decade or so. Due to its popularity, last night there were two services, and we chose the early service so we could spend the rest of the evening in last minute chores. Christmas Eve at our house is steeped in tradition. There’s fried rice. Originally that was a comfort food prepared by Linda’s mom. Mark has taken over that task these last six years, but last night we dispensed with that since it was only Linda and I. Mark was at home doing his own last minute chores.

We rushed to get to the stores before they closed, but things have changed a bit over the years. Many stores were open until seven or even eight o’clock. We made the round of open stores getting a few, last minute essentials for the next day's festivities. We then found a Burger King that was open 24 hours during the holiday. We had a happy time with the staff wishing each other Merry Christmas, and Linda and I shared a double order of Alaskan fish sandwich. Long ago Burger King had a nice fish sandwich called the "Whaler." Today’s fast food restaurants mostly serve a tiny fried fish pellet on a bun, but Burger King still has a pretty good fish sandwich. So, after last minute frantic shopping for just a couple more gifts and some desserts for today’s feast and a light meal, Linda and I cruised around enjoying the season’s lighting.

I thought back to a Christmas many, many years ago. We were still in our first house over on Sherman Street and our kids were little. It was a Christmas eve we spent with our good friends Steve and Sandi. I don’t recall exactly, but I think our boys were probably at their grandparents and Sandi’s girls were with down in Trinidad. So it was just us four driving around on Christmas Eve.

First we had to find somewhere to eat. Back then, especially around nine at night, everything was closed. We found a Chinese Restaurant to have our evening meal. Just like Ralphie and his family in the A Christmas Story, we found an open business staffed by non-Christians. Maybe our new tradition of fried rice dates back to that Oriental experience. I don’t know.

So, later, we were driving around deserted streets and not ready to go home and go to bed. Steve and Sandi didn’t have a Christmas tree. They intended to get one, but they kept putting it off. We drove by several closed lots with many trees left. In those days, artificial trees were not as popular as today, and there were half a dozen temporary locations in Longmont that offered live trees … at least they were living just previous to being cut down.

All the lots were closed, so we schemed how we would climb the fence and rescue a tree. It wasn’t exactly a felony. We might return the next day and pay for the tree. Alas we didn’t resort to thievery. No one wanted to tie a tree to the roof that cold evening.

We eventually drifted home. We dropped off our friends and returned to our house and warm beds to await the Christmas morning.

Then there was the Christmas of the great blizzard. Our boys were already at their grandparents out in Horseshoe park, about ten miles out of town toward Berthoud. I had the yellow van back then and we still lived on Sherman. It had been snowing heavy all day. About four in the afternoon, as the sun dipped and the snow blew, we hopped in the old Dodge and headed for grandma’s. We didn’t make it to the end of the block before getting stuck. The van was not too bad in snow, but the high drifts that the city had not plowed (“only arterials during this snow emergency”) were not passible by the big yellow truck. So we reversed back up our street and got the van safely back into the garage.

We spent the remainder of the evening shoveling the continuing snow off our driveway and Linda was very sad and upset not to be with her kids on Christmas Eve. The next morning the snow had stopped and enough cars and four-wheel drives had driven down our street that we made it to Main and out on the plowed highway north of town. We turned down Bob and Bea’s road which had been shoveled by someone in a tractor, and started down the narrow path in the four foot snow to their house.

It was a total of about three blocks, and I made it for two before the snow closed in on my van and we were stuck. Due to the high snow on both sides, we couldn’t open the doors, so I climbed to the back and got out that way. We walked to Linda’s folk’s house and brought back assistance with shovels. We soon had the van unstuck and made it down their driveway for a Merry Christmas.

There were other times when the whole fam-damly was here from Alaska and on New Years day I got a speeding ticket in the van as I rushed to load up everyone for a trip to the airport.

There are other times over the last fifty or sixty years that jockey for position in my memory. Many of the events just blur together in a snow storm of present wrapping torn off in anticipation of what's inside. Happy days and happy times as — what else — happiness filled our home. We're ready once again. The tree is up and the floor is littered with wrapped gifts, although not as many as in times past. There's eggnog in the frig and fruit and nuts in the bowls awaiting eager hands. Family and friends will arrive soon, and I'm still in my bathrobe. The hot coffee and pleasant thoughts are my companion on this still morning. It isn't a white Christmas this year, but it will do.

All of these memories come rushing to the forefront of my mind as I access the index=Christmas. There’s many more tales to tell, but I’ve made a New Year’s resolution to always be terse and laconic. So I’ll stop with this simple Merry Christmas to all and to all a Good Night.

Bosch Über Alles

Warning: The following is an unpaid product endorsement.

I just can’t get over what a great job the new dishwasher did over the holiday. Since it is machinery, it is my responsibility in the kitchen. I load and run the dishwasher. I got a lot of practice over Thanksgiving. At one point, I ran three successive loads and even washed some of the heavy-duty cooking dishes. Linda said, “Are you supposed to put pans like that in the dishwasher?” I replied, “Sure, they fit!”

A few months ago we were struggling with our old dishwasher. The tops of the glasses would come out full of soap and had to be re-rinsed and dried. I called a repairman and he diagnosed the problem as a failing pump. Without full water pressure, the rinse cycle wasn’t getting all the soap out that was trapped in the small depressions on the top of glasses and cups. Plus, the cleaning just wasn’t up to snuff.

As I suspected, he said that a new pump would cost as much as a new dishwasher, so I went on the hunt for a replacement. I checked out Consumer Guide (I have a subscription for the online Consumer Guide) for the best dishwasher. I always check CG for any major purchase. I like to have the best and, often, the cheap turns out expensive. The current dishwasher is about fifteen years old, but I expect good machinery to last longer than that. Sadly they really don’t make things like they used to. Most of the latest gadgets are designed for about five years of life and then they are not only obsolescent, but most likely kaput.

I was not surprised to learn that the top three or four models of dishwasher in Consumer Guide were Bosch. I first saw that German model at a store in Beaverton, Oregon. My dad took his rug shampoo machine into this little specialty shop for repair. While there we looked at the Bosch dishwashers they carried. The owner and chief handyman gave us a great sales pitch about the quality of Bosch. I remember the particular model my dad looked at had a flat drawer on top where you would put your silverware one piece to a slot. That prevented “spooning” from hiding a dirty side of a utensile from the cleansing water.

I went down to Lowes. Both my nephew Joel and my son Mark were working there, so I got an employee discount. I picked out a very nice model of Bosch in stainless. The only choices were stainless or white. Our kitchen is almond. (That’s right, we built the house in 1986. You can tell the age of any home by the color of the appliances.)

I know from watching shows on the Home Network or the Realty Network or something like that, that everyone swoons when the kitchen appliances are stainless, so that’s what I chose. Of course, “stainless” is a misnomer as the brushed metal shows fingerprints like a police blotter. Still it does look nice, and — best of all — it does a super job of cleaning.

In the CG report they showed how they tested dishwashers with baked on peanut butter and other impossible dish tests to determine the best cleaning dishwasher. As usual, they also look at value for the dollar. Even though Bosch is no bargain basement brand, it took the top three or four spots in the evaluation.

So I bought one of the top models and had it installed. I don’t mind doing electrical work, but I leave plumbing to the professionals. I like the fact that, if the installation fails and my kitchen floods, I have a major corporation to back me up. That happened to my neighbor. Her new dishwasher ended up costing almost $6,000 when you include the new kitchen they had to put in when the water hose came off. I wanted the insurance of an installation warranty just in case. So far, no kitchen flood.

The funny thing is that Lowes insists on selling you a new water hose … the fancy metal woven, high quality, and quite expensive kind. When the installer put in the dishwasher, he said to return the fancy hose for a refund since Bosch has a permanently installed hose. And it’s just a simple plastic hose. He said that must be new. I hope that Bosch isn’t “cheaping” out on me and that hose is good for twenty-five years. One advantage of being 66 is that a twenty-five year guarantee is really a life time warranty.

Anyway, this dishwasher fulfills all promises. Linda used to prewash things before they went into the dishwasher. Her neanderthal husband just puts in the dishes and glasses, all stained and yucky. I press the button. (The dishwasher actually adjust to how dirty the dishes are and adjusts the time of wash … don’t ask me how it does that, but it has a count down clock and I’ve seen it adjust when the dishwasher was full vs. partially filled. Maybe that’s the metric it uses to set wash cycles.) It also only uses one shot of soap; not the two cups of soap … one with a cover. It also told me when it was out of “finish.”

So there you have it folks. If you want to buy the Cadillac of dishwashers (or maybe, since it is German, the Mercedes of dishwashers), then this consumer analyst highly recommends the Bosch dishwasher. Tune in tomorrow when I’ll be writing about our refrigerator. It’s the original from 1986, and it has taken a lickin’ and kept on ticken’.

Christmas: Love — Family — Faith — Tradition

While at “Mollie McGee’s” on Saturday, (a local arts and crafts show), a phrase caught my eye. I think it was on the side of a wooden box, possibly an antique container of some sort. It said, “Christmas” followed by four words: “Love, Family, Faith, and Tradition.” These four themes seems like a good way to view Christmas. Of course, even though the list included Faith, many would argue that, first off, Christ is the “reason for the season.” So that is the one theme we should focus on. That’s true, but it is also true that Christmas in American, and possibly the whole world, has become more than just a celebration of Jesus’ birth. Like much in life, that fact has a good side and a bad. I wish to explore the good.

Love

First and foremost, it is certainly about love. Not just Christmas, but every day of the year. Let’s start with John 3:16: “For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life.” Yet that quote might apply more appropriately to Easter, or — as I prefer to call it, “Resurrection Day.” Still Christmas was when Christ came to us as a mortal man and thusly “love” entered the world.

The babe in the manger grew, and, at the age of thirty, began His short ministry. The theme of love was always in his message. His words recorded by Matthew, Mark, and Luke: “He answered, ‘Love the Lord your God with all your heart and with all your soul and with all your strength and with all your mind'; and, 'Love your neighbor as yourself.’" (Luke 10:27, but also Matt 22:37 and Mark 12:30) First and foremost is to love God, but it is the second commandment for us to repeat that love to others. This was the new covenant. Rather than ten commandments, we now have just these two: to love the Lord and to love our neighbors.

But who are these neighbors? Is it just the people that live next door? No, it is much more than that. In Matthew 5:44, Jesus states, “But I say unto you, Love your enemies, bless them that curse you, do good to them that hate you, and pray for them which despitefully use you, and persecute you;” The commandment to love our neighbor has universal application. We are to love everyone, including those that don't love us, as we love ourselves and as we love the Lord God.

We’re also told in Luke 14:12-14 “He said also to the one who had invited him, ‘When you give a luncheon or a dinner, do not invite your friends or your brothers or your relatives or rich neighbors, in case they may invite you in return, and you would be repaid. But when you give a banquet, invite the poor, the crippled, the lame, and the blind. And you will be blessed, because they cannot repay you, for you will be repaid at the resurrection of the righteous.’” So that’s who we are to love: our enemies and our friends and family as well as the poor, crippled, lame, and blind. Our neighbors and our enemies. Those in our country and those abroad. This is not an exclusive list. This is an inclusive list. Simply, we are to love one another — all mankind. This is a reflection of God’s love for us.

In Matthew 25:45, Jesus ties these two commandments of love of God and love of neighbor into one explanation: “Truly I tell you, whatever you did not do for one of the least of these, you did not do for me.” So it isn’t two commandments at all. It is a single commandment that we are to love Him and to love His creation. For we express our love for God through how we treat others, neighbors and foreigners, friends and enemies, the greatest among us and the least, the poor and the downtrodden; we are to love all people.

It is very clear. It is how we treat others … including the least of those … is how we show our love for Him. That is part of the love of Christmas. It is a special time that we focus more on our neighbors and those in need. That is one explanation for the presents. Although it is sad that this Christmas “love” can’t continue throughout the year, it is still good to have a special season to remind us of His love. Think of the acts of kindness and fellowship that have become the very spirit of Christmas.

There’s all those stories about love at Christmas from Dicken’s “The Christmas Carol,” where Scrooge learns to abandon greed and avarice and to love humanity, to the William Sidney Porter’s “Gift of the Magi,” where love is shown to be sacrifice. The reason for the season is “Jesus,” but the theme of the season needs to be “love.” We are commanded to keep his word: “Jesus answered him, ‘If a man loves me, he will keep my word, and my Father will love him, and we will come to him and make our home with him. He who does not love me does not keep my words; and the word which you hear is not mine but the Father's who sent me.’” — John 14:23-24. "A new command I give you: Love one another. As I have loved you, so you must love one another.” — John 13:34. What better way to honor His birth than with love.

Since I was married shortly after Christmas, there is a special kind of love that I associate with this the season. The love between a man and a wife is an institution and a miracle instituted by God in the Garden of Eden. What the Lord has joined let no man break apart.

One thing I’ve learned as both a teacher and a parent is not to assume others know what I’m talking about. Explanations must be given in terms that can be understood. Don’t assume that the message is getting across simply because you give a speech or quote a verse. So what does it mean to love one another? How do you recognize love?

The apostle Paul explained love in very simple terms. In First Corinthians 13:4 - 7, he wrote, “Love is patient and kind; love does not envy or boast; it is not arrogant or rude. It does not insist on its own way; it is not irritable or resentful; it does not rejoice at wrongdoing, but rejoices with the truth. Love bears all things, believes all things, hopes all things, endures all things.” That makes it clear what love means and what it looks like. These are words for all the days of the year. They are also words very appropriate to any marriage vow. They are words that should be at the foundation of any celebration of Christmas.

Family

Love also is a segue to the next term, “Family.” Family is a physical representation of love. Through marriage, birth, adoption, even divorce, and just circumstance, families are formed. Christmas is a time for family. For, in the American folklore, both from by-gone centuries and modern times, Christmas is often the setting for reuniting of family. Whether it is an annual trek to grandma’s or mom’s, by car or plane; or it is the reuniting of family and prodigal sons or daughters and broken relationships healed by the season that is often the theme of sappy and soapy tales from the Lifetime network. These reunions are the plot of many a Christmas story and the cause of crowded airports in late December. Travel and “getting home for Christmas” is the story line of many a yuletide tale.

It seems the entire extended holiday season from Christmas to New Years, often a week or even two of vacation, holiday, and time off from from work, is ideally suited for travel and reunion of family. Perhaps, at least in the Northern hemisphere, the weather may not be the best for long distance trips, still we’re reminded by “Over the river and through the woods, to grandma’s house we go” that Christmas is a time to visit loved ones.

I’m quite fortunate that most my family lives here in northern Colorado. Linda’s parents, our two sons, our grandkids, they are all here. There might be a new family travel tradition starting at the home of Linda’s brother Chuck and his wife Dawn in Alaska as their family has migrated to the lower forty-eight, but has returned to North Pole at Christmas these last two years. Talk about weather related travel! If it doesn’t seem much like Christmas in California or Georgia, Alaska is sure to provide the requisite snow and burning yule logs. Roasting chestnuts seems like such a winter occupation, it is hard to get the spirit when you live amongst palm trees.

I grew up in central Montana with my parents, siblings, aunt and uncles, and maternal grandparents. We would all get together to celebrate Christmas, and no long distance travel was involved; just a drive across town. Here in Colorado, for the last nearly forty years, we too have not had to drive a long way to enjoy Christmas and family.

Although things have changed a bit around here in the last ten years. With the loss of Linda’s mom we have moved the celebration from her house to ours. That is actually a transition we started before she died. The Christmas morning celebration and the food and fellowship that follow are now hosted at our home. Although we’re blessed with two sons that both are excellent in the kitchen and although Linda and Mark labor intensely with my help, we still wonder at how easily her mom was able to pull off the celebration and meal. She had the true gift of hospitality and we miss her help in the kitchen as well as her presence in our lives.

It has also changed a bit regarding presents as we are no longer as focused on gifts under the tree as we once were. With the exception of wrapped surprises for the young ones, Christmas has changed from a pile of gifts for everyone in the gathering. At some point it just got a little too much to try to figure out exactly what each family member wanted. Children, now adults, became more difficult to shop for and we gradually de-emphasized the gift giving to focus more on the the special day. Now opening presents has become a simple treat for the kids, and not a giant burden of shopping for just the right present for all. Although I miss the mystery of a present under the tree, we have found it a simpler time of year once we’ve deemphasized the commercial and focused more on the companionship.

A friend asked me if I had a photographic memory. I wish my memory was eidetic. No, I don’t have that talent. I don’t really have a good memory at all. I have to study and study to get stubborn facts into my thick skull. On the other hand, some events from my life are preserved in living color, and I play the projections on the back of my skull often. Oddly, Christmas past has sort of blurred out of focus. My childhood memories of Christmas are not very prominent in my recollections. Nor are the Christmas years I spent in the Navy, for I didn’t bother going home at that time of year, but remained on board ship and celebrated with my shipmates. I’ll be darned if I can recall those times. They’re lost in the mist of failing recollections.

My memories of Christmas as a child are sporadic. I remember opening presents and Christmas dinners and parties at my great aunt's. I remember my grandparent’s tree. It was only four feet tall. They had a built-in buffet in their dinning room, and they would put a small tree on top of that. I remember family gatherings at the table and playing cards into the late night with my uncle and aunt. There were no trips and no adventures to really cement Christmas in my youthful reflections.

I know there were always lots of gifts when I was a child, and I’ve written about some of them in my blog. One memory that is very strong is the Christmas of the Lionel Train. For some reason I was very focused on what the gifts cost. I have no idea how I knew, but I remember the “transformer” cost $49 and the train set was the same, for a total of almost $100. That was a lot of money in the 50’s and, although we lived comfortably, my parents were not rich. It was a train set we shared with the whole family and it was very rugged and easy to use, but it was a small set.

On the following Christmas, my dad got us a pair of track switches. I remember they cost $32 and had levers on the end of a wire to switch the train remotely. Now we could build a track layout that had a little more variety. My friend, John Barr, had an HO train set that was much more realistic. But the big O gauge Lionel was perfect for kids. I wish I still had that train, but my dad ended up giving it to my mom’s uncle that had a large train collection.

One thing I do remember from childhood and from my early years as an adult is that special feeling when I wake up on Christmas morning. Like any school kid, I have trouble going to sleep on Christmas Eve as visions of sugar plums … actually don’t sound appetizing … danced in my head. As a parent, I would always wake early on Christmas morning, even before the boys, and I’d wait quietly by the tree for everyone to get up and join me for the special day. Opening the presents was the central event and a good time was had by all.

That’s kind of what family is for. Sharing and giving. The real gifts I got from my parents have lasted for all these years. That is the gift of love, self-reliance, joy of learning, and a teaching of how to love my wife and raise my own kids. I tried to practice those gifts throughout my adult life. Christmas brings back all these memories of family. It may not be the exciting event it once was to me, but it still raises the goosebumps as memories merge with events of the day and the joy of love and family combine. I’m still the first one up!

Faith

As I mentioned in the first paragraph, faith is the reason for the season, and Linda and I enjoy the Christmas programs and the focus on Jesus. In the past we were regular attendees at the Christmas play at a nearby mega church and our own church holds a Christmas Eve candlelight service that we never miss.

I remember the time that our pastor, my good friend Tom Beaman, came out dressed as Joseph and told us Mary and the baby were fine. He proceeded to tell the entire Christmas tale in the first person, concluding with all the lights turned off and one by one we passed the flame from the Advent candle down the rows until the sanctuary was brightly lit like a church from ages past. Last year our friend William joined us and our new pastor repeated the celebration of depicting the light of the world coming again. This year, we’ll be there for that special service.

For those who follow the Christian faith, the birth of Christ represents the first step in God’s fulfillment of promise and prophecy. For it is through the virgin birth of Christ that God sent His Son to suffer and die for our sins. This is the act that unites all Christians in a common belief, no matter how much they disagree on other creeds and interpretations.

It is at this time of year that we’re united with others of our faith and belief. We are taught to share our love with all, Christian and otherwise. Here in America, Christmas has become a secular religion and the sacraments are shopping and eating. That’s OK. Anything that can remind us of the season and what it really means is good and important. I don’t mind when I see Santa because he represents charity and giving. I do wish we’d reduce the focus on commerce, however. This time of year was not meant to be the season of greed and desire, but rather the season of blessings. Sometimes the music and the bustle makes it hard to contemplate the inner meaning of the holiday.

The worst thing that has happened to Christmas in our technological and jet plane age is the rushing and scurrying that has taken the place of relaxed winter thoughts. It should be during some quiet time during this hectic season that we stop and consider the power of love, of family and friends, and the gift from our Savior and Lord. This is a time for church, not a time for shopping. In that regard, we have gotten things out of proportion. The reason for the season is not to wish everyone the inclusive “Happy Holidays.” Many businesses now depend on this time of year to fill the cash registers and hire extra help to deal with the crowds. Starting with Black Friday, it is hard to tell what the true reason for this holiday is. At least all the stores close for Christmas Day. That's a small victory.

Yet the secularization of Christmas isn't all bad; for the love of Christ is inclusive. He loves all of mankind. This is not a time to exclude others that follow another faith, Christmas time is a time to celebrate Christ and love everyone. Sadly, it seems that that celebration is lost on many Christians, just as it is lost on those that don’t follow Christ. The best way to restore the true meaning of Christmas is not in a cheery “Merry Christmas,” although that is a good start; but in our attitude. Let it begin at Christmas and extend throughout the year as we show the love, love of family and love of others, to all men and women of all religions and beliefs that our Lord is the Lord of love, forgiveness, and grace. This is how we honor Christ and our faith, not just one day a year, but throughout the year.

Tradition

Finally, in what I think is the correct order, comes tradition. For Christmas has become much more than an anniversary of a bible story … an anniversary most likely held in the wrong month of the calendar. It is very unlikely, based on ancient writings and practices, that Christ was born at the end of December. That date and much that happens on Christmas is more the result of tradition developed in the two thousand years hence.

I don’t find this all together bad or wrong. Christmas is about tradition. The way we celebrate has little to do with the original setting in Bethlehem, and much to do with traditions from Europe and England. That’s OK. These are good traditions that we carry on. The decorations on the tree and the lights on the houses are part of the reminder of this special time of year. If it were not for holidays and celebrations, our lives would be entirely focused on the day-to-day routine of life. We need a season to stop and reflect and all the glitter and glamor and noise and confusion is part of the trigger to that meditation.

Although the gifts may be in honor of the tradition began by the Magi, most of the trappings of Christmas come from other traditions. From the Christmas Tree to the Yule Log to Santa himself, this was all added later. The afore mentioned tale by Dicken’s has created much of the traditions we now celebrate. But these are not the traditions dearest to my heart. No, it is the personal traditions that I hold closest.

As I previously stated, I don’t really have a lot of specific memories of my childhood Christmas times. Christmas in the 50’s was not that different from Christmas in the 40’s so well chronicled in the movie “A Christmas Story.” This tale of Ralphie and his quest for a BB gun that everyone warned would “shoot his eye out.”

The move is a 1983 Christmas film based on the short stories and semi-fictional anecdotes of author, radio personality, and raconteur Jean Shepherd. The film has become a holiday classic and is shown numerous times on television during the Christmas season on the network TBS, often in a 24-hour marathon.

In 2012, the film was selected for preservation in the National Film Registry by the Library of Congress for being "culturally, historically, or aesthetically significant.” So there you go, you don’t need your own Christmas memories. You can share Ralphie’s (or Jean’s). I think this comical tale of boyhood wishes and a loving, although clearly not a perfect family, are the memories I have in my heart, even if they are not my own.

Besides, this slightly more modern tale probably fits our current Christmas traditions better than Jimmy Stewart in “It’s a Wonderful Life.” I’ve seen that one about ten dozen times too, but I always miss the beginning where everyone is praying for poor George Bailey, the “richest man in Bedford Falls.” The movie has plenty of snow and a Christmas tree, but I never really think about it as a Christmas tale, maybe since much of the action occurs in earlier flashbacks. Not a lot of Christmas tradition in that popular Christmas tale, yet it is a good story about love and family.

One year I was teaching in New York before the holidays and got to enjoy all those metropolitan traditions from the big tree at Rockefeller Center to the window displays at Macy’s, Bloomingdales, and Saks Fifth Avenue. They even put up railings on the sidewalk to separate the window shoppers from the hurrying crowd. A visit to FAO Schwartz near Central Park completes a Christmas tour. You can also tune into Turner Classics TV network which will display a plethora of black and white Christmas memories located in New York City including Bing Crosby and “White Christmas.” It seems like the American tradition is Christmas in New York.

I love New York City during December, but I’m happy to be here in Colorado. We are the modern traditional Christmas capital of the world, you know, complete with ski resorts and sleigh rides. Of course, that may be more Aspen or Telluride than Longmont. Still, if you watch TV, you’ll see a Colorado Christmas portrayed complete with cowboy boots. Maybe it’s because I’m a westerner that I find Christmas in Colorado superior to the earlier portrayed Christmas in New York … or L.A.

Another common tradition at this time of year is Christmas Carols. I love to sing along, even if I only know the first verse. We would start to sing these ancient (and a few modern) songs at church after Thanksgiving. You would hear them in the stores and live singers would appear on street corners. I love to sing along and share the memories of generations past as these traditional melodies and lyrics fill the air this time of year. There are Christmas concerts with kids and grandkid’s school bands and symphonies of bells and other instruments to attend. We always have Christmas music here at home and I wirelessly spread the iTunes “Holiday” genre throughout the house with “random” play. An odd mix of ancient music with modern technology.

There is also a classical music tradition from "The Nutcracker" by Pyotr Ilyich Tchaikovsky to Handell's "Messiah," Christmas music has several powerful musical compositions. My dad is a talented singer and I fondly remember listening to him perform the Messiah when I was about ten years old.

For unto us a Child is born, unto us a Son is given,
and the government shall be upon His shoulder:
and His name shall be called Wonderful, Counsellor, the mighty God, the everlasting Father,
the Prince of Peace.

Then there is the big “traditional” question of “when do we open our gifts?” Since, following the Jewish calendar, Christmas actually starts at sundown on Christmas Eve, many will open their presents at that time. Since a lot of modern life is extended and broken families, several Christmas events must be held at different locations. That may be the reason that gift opening begins the night before. Then there can be more gifts on the following Christmas day and a tour of family is begun. Trust me, the kids will never complain about an early opening of gifts.

In our family tradition, despite the protests of the youngsters, we open gifts on Christmas morning. In fact, as we wait for the family to gather, it is much more like Christmas noon. We used to go to Linda’s parents on Christmas day. At a few very special times one of her brothers would be there and one joyous year in the nineties, the entire Alaska clan was present. I remember a couple of years that Linda and I actually spent Christmas eve at the Lincoln’s, sleeping in the guest bedroom. That was very special too. Again I was the first one awake Christmas morning.

After a while the traditional start of Christmas became breakfast with Bob and Bea as Linda and I awaited the arrival of the rest of the family. Linda’s mom would always make a cherry cobbler and Linda would often prepare eggs and sausage. Somewhere along the line we got the tradition of fried rice the night before Christmas. Bea originated the recipe, but we still do that most Christmas Eve’s in her honor. How many Christmas traditions are based on a specific meal or food item I wonder.

In those early days we would have a big turkey dinner after the presents, and I always enjoyed the evening snack of turkey sandwiches. We’ve changed the menu a bit over the years, and last year we had a cold cut buffet so people could just graze when the urge came over them. That was very successful. After all, turkey again, so soon after Thanksgiving, seems to lack originality. It also eased both the cooking and clean-up chores and allows everyone to participate in the gathering.

IBM would provide events and funding for department parties and other gatherings back when I still worked for a living. One year, early in December, our department went to a cooking school and prepared a gourmet meal that we then enjoyed as a team building event. The main course was a prime rib roast. I liked it so well I decided to cook it for Christmas. The sticker shock for that particular cut of meat was a bit stiff. As I recall, I spent over a hundred dollars on the beef. Now I understand why everyone cooks turkey.

When I got back to work in January, my boss said he had done the same thing only his roast cost nearly two hundred dollars. We both agreed it was a very nice meal, but we would go back to turkey the next Christmas purely for financial reasons. Turkey, even at a dollar a pound, beats prime beef cuts at around fifteen dollars a pound!

Another tradition is decorating the house, a Christmas tree, and driving around to see the lights in the neighborhoods. We actually start the celebration on the Friday after Thanksgiving with our annual trek to Estes Park for the Parade of Lights. It is usually a cold night, but we bundle up and make our pilgrimage to our summer vacation spot. I fondly recall taking Alyssa up there when she was just knee high. One year we had been waiting for about two hours for the parade to start, standing around in the cold. (You have to get there early to get a parking place and a good spot on the parade route.) Just as the parade started, Alyssa said she was cold and had to go inside.

So we retreated to “Penelope's,” a local hamburger joint. We ended up getting a table right in the window and watched the parade from the warmth of the restaurant. Now that’s a tradition I can live with. Last year the night was unseasonably warm. I could get used to that too.

As I’ve grown old and gray, I’ve joined personal traditions with national and international. I think now that it is the traditions more than the gifts that are now celebrated in my heart. A time for love and faith, family and traditions. This is the most special time of year. The simple repetition of habits and music and menus has a familiar and comforting effect. The joy of the children as they anticipate the presents mingles with my memories of Christmases past. It is nice to recreate those memories each year.

While all four: love, faith, family, and tradition are very important components of the season, I think the last is greatest in my heart. For Christmas reminds me of the past. The ancient past in that town of Bethlehem, as well as my childhood and the Christmases enjoyed when my family was young. Christmas before we lost Linda’s mom and Christmas before our boys grew to men. Christmas reminds me of gifts, and the great gifts I’ve received during my lifetime. I remember Christmas in Lewistown and Christmas in Longmont. So here it comes again. The sun having completed its annual trek around Sol, it is time for love, faith, family, and tradition.

You can capture a whole philosophy in a single word — if it is the right word. “And now these three remain: faith, hope and love. But the greatest of these is love.” — 1 Cor 13:13.

Beef Burgundy

Here’s a tale I’ve never told: Beef Burgundy. When I was in the Navy and living at 8240 McCloy Road in Norfolk, I always had two roommates. The people changed a bit over the about four years I lived there as people got out of the Navy or transferred, but we were always a very close group. We were friends first and roommates second. We had a bank account for the “house” and we each payed into that account each month. If I recall the numbers correctly, our rent was $100 a month (those WERE the good old days) and we each put in $100, so we had a budget of $300 a month. After paying rent and utilities, we had over a hundred bucks for groceries.

When I cooked I liked to make casseroles. They make an easy, one dish dinner, and everything was done at once since there was only one dish. I had a “Campbell’s Soup” cookbook and, naturally, every recipe was based on their soup. There was a meatloaf that had Tomato soup and a tuna casserole with Cream of Mushroom. But the pièce de résistance was my Beef Burgundy. It used two cans of soup, Beef Broth and something else that I don’t recall now.

It was a complicated recipe. You would slice the beef into cubes. You would fry some bacon and then brown the beef in the bacon fat. It had small potatoes, carrots, onions, and garlic, plus maybe some celery. Ultimately the whole mixture went into a large casserole dish and was baked for some time. I think it served about 12. That was OK, as left-overs were another thing that three single guys would appreciate. Plus, sometimes, we had guests.

I think the dish was quite popular with my roommates, but I’ll wait for some of them to comment and respond in their own words … assuming any of them remember the dish. But this is the story I really want to tell:

I was home on leave … or it could even be after I got out of the service. Anyway, I was going to make the dish for my mom and dad. I had it all prepared and in the oven cooking when my dad commented that it smelled very garlicky. I explained that it had garlic in it, two cloves. My dad picked up the little empty cardboard box from the counter that had contained the garlics. He said, “Where’s the rest of the garlic.” I said, “I put both of the cloves into the dish.” He then explained that the little box had contained two bunches of garlic and that each bunch had about a dozen cloves. He was a grocer and an excellent cook himself, so he knew the difference between a bunch of garlic and a clove. Apparently I didn’t. Well, I was only twenty-something years old and not a great cook … obviously, so I didn’t know the difference. I had put something like 20 or 30 cloves into a recipe calling for only 2.

The good news is that the extra garlic seamed to just cook out of the dish and, other than tasting very, very Italian, the Beef Burgundy was OK. At least it was all eaten up. Just as a good airplane landing is any that you walk away from, a good dish is any that people will actually eat.

So that’s my story. I haven’t made Beef Burgundy since then … it is a lot of work. But I have recreated some of my other dishes from the “Campbell’s Soup Cook Book.” Come over some time when I’m cooking and check for garlic in the air.

Sanding Away the Old

During the last three weeks I’ve had the opportunity to use several techniques, materials, and processes from the “home improvement” inventory. Linda and I have been helping our friends Steve and Sandi work on Sandi’s mother’s house preparing it for sale. Her mom passed away a couple of months ago and most people my age have experienced the issue of preparing the “parent’s home” for sale.

My father was and is an excellent craftsman. He built our first home in Lewistown, and I mean “built” as it was with his own hands. He’s always been a craftsman and an expert at many building trades including installing tile and building cabinets. If I had inherited only 1% of his skills in this area, I could have my own home improvement show. I did learn a few things from him as a child while I was busy losing his tools. I’ve had to refresh several of those memories and skills these last few weeks.

I’ve never done much painting, unless you count the motorcycles we refinished back forty years ago here in this very area of Tidewater, Virginia. With a son who is a skilled professional painter, I’ve always contracted even the simplest paint jobs to him. This time he wasn’t here to help, so I spent some time behind the brush and roller, but mostly I worked on other tasks.

There was considerable electrical work, installation of new switches, outlets, and even lights to be installed. I replaced a couple of electrical boxes and even did a little rewiring. I put new locks on the front door and painted the handle to match the new hardware. There was some plastering and spackling to be done to close all the holes not covered by switch and receptacle plates or to hide where curtains once hung. Mirrors and cabinets needed hanging in bathrooms and the living room and a large bay window had a shelf with water damage that I completely refinished in a three day orgy of sanding and applying new polyurethane. I’ve done so much sanding that I think my arms are going to fall off.

The last four days were spent refinishing the kitchen cabinets, although I haven’t completed that job. We leave tomorrow, so that task will be left unfinished for Steve and Sandi to complete. I’ve prepared all the cabinet faces and most of the doors and drawers are ready, although there is a little more detail sanding to be done on some doors. I bought 36 new nickel plated knobs for the cabinets, and I’m sad I will not see the finished project.

A lot of the sanding was by hand … I also refinished seven window sills. I resorted to power tools for the bulk of the cabinet refinishing. You should see me with my protective ear covers, dust mask, and googles. I look like an alien visiting the Earth. Plus, all that stuff is hot and I sweat a lot just watching an exercise program on TV, so you should see me when I actually do physical work.

Still, there is little to compare to the satisfaction of taking something worn and old and making it look fresh and new. There’s still plenty to do, and some choices will be made by budget restrictions. For example, there are louvered doors into the pantry that are a dark and ancient finish. I’d love to replace them with brand new doors freshly sprayed a nice white color. Ditto for the dark trim on the kitchen doorway. Whether Steve and Sandi will do that remains to be seen. Still it is satisfying to see the rebirth and restoration that a little sandpaper and sweat can do to wood … nature’s purest product. No plastic, steel or glass can equal the warmth and comfort of wood. I prefer natural finish, but paint is good too and can cover a wealth of wear. Sort of like makeup for the soul.

One can make comparisons to life and how we are gouged and scratched by the “slings and arrows of outraged fortune.” We could all use a little sanding, wood putty, and refinishing as we grow older. (Plus a little hair restoration … but I digress.)

I keep thinking philosophically while growing callouss on my hands. The ancient Greeks were great thinkers, but they eschewed physical labor. That’s their loss. The mind works better when the hands are busy.

I’ve been busy thinking while the repetitive and monotonous work is being done. Hard work is good for the mind as well as for the soul and the body. I’ve even come up with some new insights into physics during the repetitive, Karate Kid, “wax on, wax off.” I think the path to deeper understanding of quantum physics, relativity, and string theory lies in the basics of classical, Newtonian understanding. A good house, and a good finish, must have a good foundation. Preparation is the key to the final coat being smooth and glossy.

If I get lost in the manifold twisting of 29 dimensions of rolled-up-space, it may be because I don’t know the simple field of F = mA. I spent one whole day sanding and listening to “the greatest hits of the ‘60s, ’70’s, and 80’s,” and thinking about these basic concepts. (I had no choice but to think about basic concepts, since my grasp of the advanced subjects is lacking … after all, that’s the issue … isn’t it?)

The object of my “a-ha” was Lagrangian Mechanics. The "Lagrangian formulation" of Newtonian mechanics is based on an alternate form of Newton's laws which is applicable in cases where the forces are conservative. (Note that, in nonclassical physics, the forces may not be conservative. They may be “liberal.” — just a little political humor. ) Lagrangian mechanics adds no new "semantics" to Newton — it's just a mathematical change, not a change in the physics.

You see, Newtonian mechanics has a problem: It works very nicely in Cartesian coordinates, but it's difficult to switch to a different coordinate system. Something as simple as changing to polar coordinates is cumbersome; finding the equations of motion of a particle acting under a "central force" in polar coordinates is tedious. The Lagrangian formulation, in contrast, is independent of the coordinates, and the equations of motion for a non-Cartesian coordinate system can typically be found immediately using it. That's (most of) the point in "Lagrangian mechanics".

Before we go on I should hasten to add that the Lagrangian formulation also generalizes very nicely to handle situations which are outside the realm of basic Newtonian mechanics, including electromagnetism and relativity. I actually was thinking about the most fundamental equation of all time … at least in my opinion. That is Pythagorus’ famous equation for a right triangle. You know, the one about squares and hypotenuses. It is at the foundation of coordinate systems and measuring the curvature of non-Euclidean, Riemann space that is the basis of the General Theory of Relativity. If you have trouble understanding the complex stuff, it can often be traced to a root cause of failure to really, REALLY understanding the simple, basic, fundamental stuff it is all based on.

That was my epiphany as I repeatedly rubbed the wood, always with the grain, into its natural basic beauty found underneath all the dirt and worn finish from sixty years of wear and tear. They built houses well back in the fifties. Time and wear has added scars and wounds, but the underlying structure is still sound. It just takes a little elbow grease and 120 grit sandpaper, plus — maybe — a little help from electricity and Black and Decker.

I am going to miss the hard work and transformational effort, but I won’t miss the sore back and arms that hang helpless at my side from exhaustion. I’ll heal. The house will be beautiful. Happy memories will merge with fresh paint and polyurethane to start new memories for the next home owners. And my new insights will be applied as I get back into study mode. It is hard work to renovate a 60 or 70 year-old home. It’s much harder work to stuff knowledge and concepts into a 66 year-old brain. I need a Black and Decker brain sander.

That was the year that was!

It was a very good year.

Germany surrendered on April 29, 1945 and Japan gave up on August 15. It took a while for the boys to get home. Some were already married. Many, like my dad, married shortly after returning safe. That made 1947 the first year of the baby boom. Oh, sure, a few were born in ’46 by those that had a head start, but, with nature's delay of nine months, it is 1947 when the action started. That is the real start of the bubble that passed through the demographics like a pig being digested by a boa constrictor.

Nineteen Forty-seven. That’s the year that was. It was a very good year. And just who might be a famous person born in ’47, you ask?

Well, there’s Arnold Schwarzenegger, actor, governor, body builder, and general tough guy. Although not born on American shores, he’s a ’47 baby. Also Kevin Klien, Jonathan Banks, and James Woods. If you’re into scary movies, there’s Robert Englund and Stephen King. Also Peter Weller from Robocop and Sam Neill from Jurasaic Park. Richard Dreyfuss and Glenn Close, not exactly character actores. Then there’s Alan Thicke and Ted Danson as well as Farrah Fawcett and David Bowie. My favorite is Edward James Olmos, but also Teri Garr (remember Young Frankenstein?) and Rob Reiner — Meathead. Oh, and also Meat Loaf. There’s Anne Archer, Stephen Collins, Joe Mantegna, Larry David (Seinfeld guru) and Jane Curtin. Remember Michael Gross … Family Ties and Tremors? Well, Merideth Baxter too. Cindy Williams and Tom Clancy; Barbara Bach and Wes Studi; Kim Darby and William Atherton; Sally Struthers; Elton John and Iggy Pop. Ben Cross and David Mamet as well as John Larroquette and Takeshi Kitano … 1947.

Richard Lewis, James Hunt, David Letterman, Kareem Abdul-Jabar and Bill Smitrovich as well as Harry Reems were born in that very good year. Cheryl Tiegs and Jimmi Walker as well as O.J. Simpson, Karen Valentine, Martin Ferrero and Jameson Parker. Don’t forget Paula Dean or Sammy Hagar and Jill Elkenberry or Don Henley and Joe Walsh. Also, Gregg Allman and Michael Flynn. I’m just getting started. There’s Carlos Santana, Hillary Rodham Clinton and Salman Rushdie. Remember Jon “Bowzer” Bauman?

My wife liked Danielle Steel and I liked John Dykstra who did the special effects in Star Wars. There’s Ron Wood from the Stones and Mitt Romney from the … well … Then you can add Ian Anderson from Jethro Tull and Bill Lancaster who wrote The Thing. The lovely Emmylou Harris was born in 1947 as was Warren Zevon and Ry Cooder. Include George S. Clinton who wrote the sound track for Austin Powers and Bill Hayes who wrote Midnight Express. Arlo Guthrie, “The NY Express way is closed, man.” and Judge Joe Brown from daytime TV. Both Melanie and John Stossel as well as Johnny Bench and P.J. O’Rourke from that very good year.

Comedian and columnist Dave Berry and Jim Plunkett as well as Steve Forbes and Peter Waterman … also ’47. Who remembers Chip Moody from talk radio or Dan Quayle form … again … talk radio. Or the WCW “Giant Haystacks.” (I’m not making this stuff up, folks.) How about “Lacy Legends,” billed as a “busty” actress.

Then there’s also Ron Fleming, Jack Barney, John Barr, Gary Hornseth, Steve Miller, and Gary Murphy. Oh, and there’s also Mickey Cheatham.

Album Art

It is the nature of us folks of slightly advanced age: the Medicare Generation, the Baby Boomers, Old Farts, Former Hippies, and other general terms for anyone over fifty, to compare our times to today in such a way that it is obvious that civilization has gone to h-e-double-toothpicks in a shopping cart. We had it so good in our day … flushing toilets and the electric-light-bulbs. We rode our bikes with no helmet, drove our cars with no seat belts, and had to call up our girl-fiends (or boy-friends) from pay phones or suffer the possibility that our younger siblings would listen in on the extension.

Yup, it was a simpler time of black and white TV and AM radio. We gradually graduated from LPs to 8-tracks to cassettes, improving portability along the way, but nothing like today’s iPods and smartphones that seem to be surgically attached to every runner, walker, and stroller we see on the street and in the stores and restaurants.

Let’s talk about LPs. That’s Long Playing … as in “Long Playing Records” or, as we quaintly called them in comparison to old picture books: “albums.” Of course, we also had “45’s,” but it was album rock that really set the stage for the sixties. Soon we were consuming concept albums and enjoying the art work as we shared the music and the experience.

Now days we have CDs. Never mind that they don’t match the LP in sonic capability, what about the art work? You need a magnifying glass to really enjoy what was once a great part of the hip scene. And don’t get me started on downloads and iTunes. Sure, Apple has some of the album artwork and on a large, hi-def screen it does approach the intimacy of an album cover, but you don’t pass the laptop around the room as you listen to the music and an iPod is a terrible visual tool and isn’t about sharing at all, unless you count giving one of the ear plugs to your buddy to enjoy the left channel. And albums had more than just a front cover. I spent many an enjoyable listening hour accompanied by liner notes and intimate inner art.

Oldsters and geriatric hipsters will wax poetical about sitting around the listening room (that’s LR in architectural parlance) and passing the album artwork around while listening and possibly passing around something else.

Whether it was the special covers of “Yes” or “Santana” or the “Beatles,” these were works of art worthy of the sonic art contained on the black disks. Some were very expensive productions. I remember a Stones cover that was one of those 3-D things that changed when you moved it or a CSN cover that was almost leather in duplication of a picture album. Later, these were changed to simple photographs of the more expensive materials to save on production costs, but still the artwork was prominent.

And it wasn’t just limited to the front cover. It extended to the back, and the sleeve, and often the covers opened like a book providing two more surfaces (sometimes even more) for graphics and photos and liner notes. These are all things long gone in this age of downloaded and digitally compressed musical product that is about as tasty as a tomato shipped from Chili to us hipsters used to tasting the music fresh from the garden.

These covers were real art: Joan Miro, Max Ernst, Mondrian, Picasso, Dali, John Martin, Rene Magritte, Jean Tinguey, Bruce Pennington, Eddie Jones, Jack or Josh Kirby, and … of course … Derek Riggs.

A band’s first contact with the public is the picture on the front of the cover. This was before MTV … and I mean back when MTV was “Music” TV and before late night concerts and Saturday Night Live. You knew the band personally from the cover. Sure, there were also posters for concerts and clubs, but that’s pretty much gone too now days. A record release was a cultural event. It was a big canvas and had a big art to go with it … and then there were the “gatefold” covers that had inner surfaces and there were included notes and other goodies and even the record sleeves could receive art to complete the concept. In those days a album included a lot of art, both sonic and visual, to excite and entertain and inform the listening public. It was part of the art and the experience. As Jimi would ask, “Are you experienced?” Well … I am.

That all ended with the CD crystal case. In marketing terms the crystal case was just a complete disaster. The covers went from being impressive objects of art to practically insignificant additions. Reduced cover pictures were all there was, and they were singularly unimpressive like miniature ceramic birds.

The rise of the MP3 download has finished off the art form. Now there’s practically no avenue for you to advertise your new album and link it with a certain image or style. Your music has no covers. There is no way to link merchandise with these products; this is an important source of revenue for all bands. The customer cannot identify the product by the image or link himself to that product with a certain style. T-shirts and other tour merchandise substitute, but not at the level of exposure of the original record cover.

Yup, in my day we had it good. Sure we had to walk to school … through the snow … up hill … both ways. But we had album covers. You young whipper snappers don’t know what you’re missing. Now, about hip-hop music and rap!! Oh, don’t get me started.

Scalars, Vectors, and Tensors -- Reloaded

In 300 B.C., Euclid created (or some would say documented) Geometry as a deductive, axiomatic system. In 1637 René Descartes introduced the coordinate system in Euclidean Geometry, combining algebra and geometry into a single mathematical structure. A logical procession of this line of thinking ultimately led to the development of the Tensor.

One can say that tensors came of age with the appearance of the remarkable paper by the famous mathematicians Ricci and Levi-Civita called Methods de Calcul diferential absolu et leurs applications, Mathematische Annalen, 1901. But it is primarily due to Einstein's use of tensors in his General Theory of Relativity that was responsible for the sudden emergence of the tensor calculus as a popular field of mathematical activity.

Tensor calculus is concerned with the study of abstract objects, called tensors, whose properties are independent of the reference frames used to describe the objects. A tensor is represented in a particular reference frame by a set of functions, termed its components, just as a vector is determined in a given reference frame by a set of components and if a new coordinate system is introduced, the same tensor is determined by a new set of components and the new components are related to the old ones, in a different way known as the covariant or covariant way.

Since tensor analysis deals with entities and properties that are independent of the choice of reference frames it forms as an ideal tool for the study of natural laws. In particular, Einstein found it an excellent tool for the presentation of his General Theory of Relativity, and he spent a couple of years in correspondence with Levi-Civita to get the math right. (We all need a little help from our friends.) As a result tensor calculus came into general prominence and in now invaluable in its applications to most branches of Theoretical Physics, it is also indispensable in the Differential Geometry of curves and surfaces in Euclidean space.

Tensor is a generalization of the term "vector" and tensor calculus is a generalization of vector calculus.

Reference frames are important in Newtonian physics as well as Special Relativity, but it is the distortion and curving of space that Einstein's General Theory predicts that make tensors such important tools.

In mathematics, a manifold is a topological space that near each point resembles Euclidean space. More precisely, each point of an n-dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension n. Lines and circles, but not figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot.

Although near each point, a manifold resembles Euclidean space, globally a manifold might not. For example, the surface of the sphere is not a Euclidean space, but in a region it can be charted by means of geographic maps: map projections of the region into the Euclidean plane. When a region appears in two neighboring maps (in the context of manifolds they are called charts), the two representations do not coincide exactly and a transformation is needed to pass from one to the other, called a transition map.

The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows more complicated structures to be described and understood in terms of the relatively well-understood properties of Euclidean space. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. Manifolds may have additional features. One important class of manifolds is the class of differentiable manifolds. This differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model space-time in general relativity.

As a kid interested in cars and souped up cars called "hot rods," I find a good example of a manifold in the metal parts of an engine that share the term. Both the intake manifold and the exhaust manifold are good examples of the twists and turns that are part of the geometric representation of mathematical manifolds.

The notion of a fiber bundle first arose out of questions posed in the 1930s on the topology and geometry of manifolds. By the year 1950 the definition of fiber bundle had been clearly formulated, the homotopy classification of fiber bundles achieved, and the theory of characteristic classes of fiber bundles developed by several mathematicians, Chern, Pontrjagin, Stiefel, and Whitney. Steenrod's book, which appeared in 1950, gave a coherent treatment of the subject up to that time.

Around 1955 Milnor gave a construction of a universal fiber bundle for any topological group. At that same time, Hirzebruch clarified the notion of characteristic class and used it to prove a general Riemann-Roch theorem for algebraic varieties. This work led to methods for mappings between various manifolds and ultimately led to a mathematics course that combined Fiber Bundles and Cobordism with topics of principal bundles, vector bundles, classifying space, connections on bundles, curvature, topology of gauge groups and gauge equivalence classes of connections, characteristic classes and K-theory, including Bott periodicity, algebraic K-theory, and indices of elliptic operators, spectral sequences of Atiyah-Hirzebruch, Serre, and Adams, Cobordism theory, Pontryagin-Thom theorem, and calculation of unorientated and complex Cobordism. I am now the freshest graduate of that class, and ready to map the reference frames until my slide-rule melts from the friction.

Now I'm ready to tackle that General Theory. I don't have Levi-Cevita to help me, but I do have my trusty Dale Husemoller "Fibre Bundles" text, even if the spelling is British. Good old Springer-Verlag, New York, Heidelberg, and Berlin. Das ist gut.

Scalars, Vectors, and Tensors

A student’s first physics class, often encountered in High School, will usually begin with a discussion of “vectors.” We assume these students are already schooled in simple mathematics such as arithmetic and geometry and even some trigonometry. They are quite familiar with the Pythagorean Theorem which gives values for the sides of a right triangle with the famous square of the hypotenuse is equal to the sum of the squares of the other two sides. I suspect all students, down through the ages, good student or bad student, it doesn’t matter, all students know this basic formula which was discovered by all the ancient cultures that existed before the modern Christian era.

For some reason, I don’t recall what it was. It may have been due to a broken bone or possibly a trip I took with my parents. In any case, I missed the first two weeks of Physics at Fergus County High School in my Junior Year. So when I showed up for class, all the other students knew about vectors and how to manipulate them, often using the ancient Greek theorem I described in the previous paragraph. But I didn’t!

While the rest of the class was learning how to add forces and manipulate the press of gravity and pushes and weights and measures, I was scrambling to add vectors to my sphere of understanding. It was a traumatic few weeks until I got caught up.

Now those who have taken the class know that things were divided into two categories or groups. One was called “scalars.” That is, measurements or values or numbers that had no direction. Scalars are physical measurements without any direction to them. For example, temperature, mass, and bank account balance. They can have a positive or a negative value, but, otherwise, they are simple integer or real numbers (decimal points).

In contrast, vectors have magnitude like a scalar, but also direction. They are like little pointy arrows or the hands on a clock face. They have a magnitude, often represented by the length of the vector in a geometrical drawing. But they also have direction like North or South, or like the previously stated hands on a clock, pointed at 12 noon or at 6.

In the physics class they could be represented by an angle from a baseline, which was usually horizontal like the floor or table top. So a vector pointing to the right, by convention, would be zero degrees and, pointing straight up, ninety degrees. If the vector points to the left it is 180 degrees, and so forth.

Then the fun begins. You could use a coordinate system such the common Cartesian coordinates, named after René Descartes, that have an “x” axis and a “y” axis … at least in two dimensions.

Now you could describe a vector as a set of numbers or coordinates. It was convenient to put the start of the arrow at the coordinate system origin, a location with x and y values of 0,0. Then you would describe the pointy end of the arrow with some value such as 2,2. That’s an arrow from 0,0 to 2,2. Some quick calculation would show the arrow has a length of the square root of (2)² + (2)², which is the square root of 8 or 2√ 2 , approximately 2.8.

Remember?

Of course, vectors can be in three dimensions too. A similar vector from 0,0,0 to 2,2,2 has a length of the square root of the three coordinates. (The Pythagorian theorem can be expanded to multiple dimensions.) So now the length would be √ (2)² + (2)² + (2)²  , almost 3.5.

Now suppose you expand it to more than three dimensions. Now it gets hard to visualize. Assuming that the vector starts at the origin, you can state a vector as a matrix with one row (or, more common, one column). For example, our vector on the flat plane would be [2,2] and if it was in five dimensions, it would be [2,2,2,2,2], where the square brackets are the symbol for a matrix or collection of numbers. And you would take the sum of the squares of the five numbers to calculate the length of the vector in 5-space.

As I pursued advanced mathematics, I learned to use matrices as a mathematical tool and coordinate systems could be represented by a matrix. Vectors and matrices are the common mathematical topics in all branches of modern physics.

Now I’m learning “Tensors.” What is a tensor, you ask? Well that’s an interesting question. They are related to scalars and vectors. They can describe the relationships between scalars and vectors and even between other tensors. Tensors are often represented by matrices with several rows and columns.

Because they express a relationship between vectors, tensors themselves must be independent of a particular choice of coordinate system. That is one of their powerful capabilities and it is why Einstein chose tensors to represent space, which he supposed was not flat or Euclidean, but rather curved by the distortion of gravity and energy. That is the basis of his general relativity theory.

Both the concept of matrices, which were used to describe basic quantum theories, and tensors being used in the early 1920’s were new mathematics to physicists. At that time they were esoteric structures only studied by mathematicians. Now days, they are part of the basic math curriculum studied by every physics student.

Einstein actually had to seek help from mathematicians as he perfected his general theory field equations. General relativity is formulated completely in the language of tensors. Einstein had learned about them, with great difficulty, from the geometer Marcel Grossmann. Tullio Levi-Civita then initiated a correspondence with Einstein to correct mistakes Einstein had made in his use of tensor analysis. The correspondence lasted 1915–17, and was characterized by mutual respect.

(Levi-Civita was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made significant contributions in other areas. He was a pupil of Gregorio Ricci-Curbastro, the inventor of tensor calculus.)

So next time you are struggling with your school homework and asking for help, isn’t it comforting to know that even the great Einstein had to find a tutor to help him? As we always said at work, “It’s not what you know, but who you know.”

Apple's New iOS and User Interface Design Principles

I was thinking about the new iOS "flat look" and about product branding. That's really the point. Oh, there might be a small amount of usability or evolution of the computer suaveness of iOS users in the transition from a skeuomorphic design. Skeuomorph are objects that retain ornamental design cues to structures that were necessary in the original. For example, making an address book look like a book or a calendar icon look like a desk calendar.

In the early days of GUI design, skeuomorphic design was considered a good visual clue for novice users and often involved a "3D" look. Now Apple has gone to a "flat look" that some find more modern and pleasing to the eye, although many have complained that the new iPhone and iPad look with thinner fonts and other changes is harder to see and use. Of course, any change to a familiar item will raise doubts and concerns in users used to the old way. That’s human nature.

You know, some design changes are like the width of ties. They just change it to force you to buy new ones. I'm not sure what I think about the new look. It is cleaner and even more modern looking, but I didn't mind the old look. Appearance of a device is an important part of the overall user experience, and I like products designed by designers rather than just engineers — something Apple was very successful with in the past. So are these changes just changes for changes sake, or are they really useful evolution of an interface used by everyone from quantum physicists to my great aunt (93 years old)? Now that is the question.

Let me change the subject slightly and speak of branding. After all, the appearance of the user interface on a product that consists primarily of just a screen … be it a smart phone or tablet … is more than just a vehicle for data entry and output, but really the look and feel is part of the branding.

Years ago, when I worked for IBM Printing Systems Division, we had a brand for our printers: “InfoPrint.” That’s a neat name. After all, IBM was in the information business, where “information” was the new buzz word replacing “data” in the marketing consciousness. (There is a difference between “information” and “data,” with the former considered higher on the intellectual scale and therefore more useful to business customers.) Over the years computers had evolved from simple data processors to “information appliances.” Anyway, InfoPrint was a pretty good name for our printers which could print text and graphics and color and barcodes and all kinds of marks on paper. Actually, we did a lot more than marks on paper. Our systems could also fax and email and even push data to web sites. Our printers could even write to RFID tags. A lot more than the conventional understanding of the word "print."

So, at one point, our gallant leaders decided we would change the “P” in InfoPrint to a lower case letter to de-emphasize printing as we did a lot more with “Info.” So it was decreed in the PSD land that the “P” would become “p.” That involved several million lines of code that had to be scanned and have the term changed on everything from computer screen output to print output to even comments and statements in the code.

Let me explain briefly how computer programs work with words or text. It is usually called “strings.” That is, strings of letters like pearls on a necklace. The common computer data structure used is an array. Now for you non programmers, a computer “array” is like a mathematical “matrix.” And for you non-mathematicians, it is sort of like a crossword puzzle. At least a two-dimensional array or matrix is like a crossword puzzle. There are rows and columns and little squares each of which fits one letter or character.

These little squares are called “cells” and you can often identify them by a number. The first cell in the row might be called C₁ and the second cell C₂ and the twelfth C₁₂. A string is usually a one dimensional array like one row in a crossword puzzle.

Now there are variable strings that store data that changes. For example, when you enter a userid or a password on a computer screen, what you type may be stored in a variable string. There are also “literal strings,” which are strings that don’t change. The programmer gives them a value and that is kept throughout the program. The literal string may have a title or name or identifier.

So our software was full of something like this: name = “InfoPrint”

Now it doesn’t seem hard to use a search and replace function to change all “InfoPrint” to “Infoprint,” but it was a bit more complicated. You see, any change to a computer program, especially a large program with over one million lines of code, which is what we had a bunch of, is dangerous. Even though this is about the smallest change you can imagine, the string length didn’t change, and it is just a literal string, so it should not have any dire consequences, but programs are very brittle and the smallest change can break something.

So even this little change required hundreds of hours of programmer time to make the changes, test the changes, verify the changes, and even fix a few things broken by the change. For example, the small “p” is narrower than the large “P,” so it can change alignment and line breaks.

So, we made the change as directed by the executives and, a few years later, all our software said “Infoprint.”

Then we were sold to Ricoh. IBM sold the Printing Systems Division, its 3,000 employees, all its patents and products, to a Japanese printer company called “Ricoh.” We needed a new name and brand since we could no longer be “IBM.” IBM would not sell the right to use that name. So, our new President (who used to be our division “General Manager” — his title was changed too) announced we would be “InfoPrint.” That’s right … the big “P” was back.

He said he had seen an old poster and knew that used to be our brand. Some snickered and asked him why. He said, “ ‘Infoprint’ is a word, ‘InfoPrint’ is a brand.” Oh. Alright everybody, back to your terminals. We’ve got some code to change.

So I am sensitive to branding and user interface changes and I’m also appreciative of design, especially user interface design … the “look and feel.” Is the new flat look more modern? Or is it just copying Microsoft’s new look and feel?

One issue with an interface is consistency. If the look is to be “flat,” then the entire look should be flat. Get rid of a “Notes” app that looks like yellow lined paper. Make the calculator look less like a business machine. Make the calendar look less like a paper calendar. Make an address book look … you get the idea. Consistency.

That means that, not only does Apple have to change the look of its OS, but also the look of all the apps. And most of the apps are not written by Apple. The change is under way. That’s one reason you see so many app updates.

A lot of work went into making the transition by both Apple and now by the creators of the popular apps on iOS. Would all that effort have been better spent on things like improving security or communications rather than simple look and feel items? That’s the question too.

I’ll conclude with a description of one of the nicest user interfaces I encountered in my early years of programming. It was a program from Lotus before IBM bought Lotus. It was called the “Organizer” and was a complete address book, calendar, to-do, etc. It was modeled almost perfectly after a typical portfolio or address book such as the Day-Timer or Franklin Planner. It looked like a book and the pages turned like a book, reminiscent of what Apple would do later. It had tabs like a book and even a cover.

Organizer also had a page for just “notes,” similar to the function of the Evernote app on a Mac. It also had a section of anniversaries that you could record birthdays and wedding anniversaries and it would pop them onto the calendar every year so you would never miss a celebration. I added the year to the anniversaries to keep track of the age of nieces, nephews, and parents. A very useful addition to any calendar. I don’t know if either Microsoft Outlook or Apple apps support that feature. I’ve never found it.

One interesting visual feature of Organizer was when you drug something to the trash bin, it would explode in fire and be incinerated. From a user interface design, although that was quire interesting visually and even had a “burning sound,” in fact the deleted item was not destroyed. You could restore deleted items if you wished. That's a misuse of a visual clue. Although it was pretty to see the wastebasket contents burned up, it implies that they are gone. Notice that both Apple and Microsoft's wastebaskets show the contents until you "empty" the basket and permanently delete the data. That is matching the interface action to the true program function. Pretty is nice, but accurate representations is best.

It was pretty and interesting and fun to use and — most important — it worked well and was easy to use. I used it during my entire time when Windows was my primary personal operating system. When I converted to Mac for personal computers in 2010, I switched to the MacOS apps of address book and calendar. Besides there being less visually exciting or downright “pretty,” they lacked important function. For example, in Lotus Organizer, I could link items together like related address book entries or calendar reminders with other data including a simple notebook page. This multiple linking was a superior function to anything else out there from Microsoft Outlook to anything on the Mac. I was even able to sync Organizer to my Palm Pilots and Palm smartphone. But, when I went to Apple for phone and computer, I was forced to change programs.

Fortunately it was easy to migrate the data, but now I’m using Address Book, Calendar, and even Evernote, where I used to just use Organizer … and they are not as pretty or “skeuomorphic.”

After reading the complaints from people about the new Apple flat look where people are finding the new font harder to read or my confusion with the “shift” key … understanding if it was pressed or not from the icon change … I’m not sure I agree with the new look.

Should Apple have invested more programmer time in new function, such as multiple links between items in various apps instead of keeping programmers up late at night changing the style of icons and screen? Well, that’s for the customers to decide … the way customers always decide … with their dollars.

Is Apple just being trendy or are they still leading the design wave? Did Apple substitute narrower ties for real functional updates? What do you think?

I think the answer will lie in what happens over the next year or two. Apple has always bragged that it doesn’t listen to customers but “leads” them. Are they leading them down a path that customers will ultimately embrace? Or is it a dead end that will boost the credibility of the competition? Time will tell?

Mathematics -- Part Four: Geometry

Geometry, literally “Earth-measurement,” is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. I’ve mentioned many times in my writing that virtually all of modern mathematics traces its roots back to ancient Greece. Geometry is the branch of math that is the deepest root. Although the Greeks did work with numbers and some other branches, it was their work with Geometry that set the pattern for an axiomatic, proof based structure.

An axiom is a statement or proposition that is regarded as being established, accepted, or self-evidently true. The Greeks thought they were the latter … “self-evidently true.” A more modern understanding is that these foundation principles are established and accepted and they are a minimum set of accepted beliefs. All else is derived from these fundamental axioms and logical techniques using “proof techniques.”

I think that all readers know that Geometry is concerned with shapes such as points, lines, circles, triangles, rectangles, etc. In addition, the Greeks had a very simple limit to the tools that were to be used to develop the discipline using so-called “constructions.” All the proofs are based on constructions that can be performed with only two tools: a compass and a straight edge. A compass can be used to draw circles and arcs and also to make certain measurements. The straight edge is like a ruler, only there are no markings. It is only used to draw straight lines and connect points. The compass is the only allowed measuring tool.

The fundamental axioms that the Greeks brought to refinement are collected in a text book called The Elements. Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the ancient Greek mathematician Euclid in Alexandria ca. 300 BC. It is a collection of definitions, postulates or axioms, propositions (theorems and constructions), and mathematical proofs of the propositions. The thirteen books cover Euclidean Geometry and the ancient Greek version of elementary number theory. Most scholars believe that the book is a collection of proofs and work developed by several predecessors, but also contains original work by Euclid. It is thought that he sometimes replaced earlier fallacious proofs with more precise and correct versions he developed.

Everything is based on the definitions and axioms. Book 1 contains Euclid's famous 10 axioms. He called the first five “postulates” and the remaining five “common notions.” One reason he divided the list is that some pertained only to Geometry and the last five were common to numbers and figures. Most controversial among the list is the fifth postulate called the "parallel postulate." Although the fifth axiom speaks about non-parallel lines intersecting, you can derive the proof of parallel lines not meeting from it. The rest of the book are the basic propositions of Geometry: the Pythagorean theorem (Proposition 47), equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" or congruent (have the same area).

The Elements was the most widely used textbook of all time, has appeared in more than 1,000 editions since printing was invented, was still found in classrooms until the twentieth century, and is thought to have sold more copies than any book other than the Bible.

This collection was so influential to mathematics for the last two centuries that new translations were being published regularly and as recently as 1939 and there are mathematicians alive today who originally learned Geometry and other math directly from this ancient book.

The Elements also contained 23 definitions for such objects as a point, a line, a straight line, a surface, an angle, a square, a circle, even the concept of the center of a circle.

Let’s examine the ten axioms that Euclid used to base his entire mathematical system upon.

1. Any two points can be joined by a straight line.

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

4. All right angles are congruent.

   (Two objects are congruent if they have the same dimensions and shape. Very loosely, you can think of it as meaning “equal,” but it has a very precise meaning, especially for complex shapes.)

5. Parallel postulate: If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

6. Things that equal the same thing also equal one another.

7. If equals are added to equals, then the wholes are equal.

8. If equals are subtracted from equals, then the remainders are equal.

9. Things that coincide with one another equal one another.

10. The whole is greater than the part.

Axioms 1 and 3 justify the two basic tools of Geometry. The second postulate gives an expression to a commonly held belief that straight lines may not terminate and that the space is unbounded. By Definition 10, an angle is right if it equals its adjacent angle. Thus the fourth postulate asserts homogeneity of the plane: in whatever directions and through whatever point two perpendicular lines are drawn, the angle they form is one and the same and is called right. We may think of the fourth postulate as having been justified by the everyday experience acquired by man in the finite, inhabited portion of the universe which is our world and extrapolated (much as the Postulate 2) to that part of the world whose existence (and infinite expense) we sense and believe in.

If this was a math class we could get into categories such as plane Geometry (figures on a flat surface) and solid Geometry. We could talk about Euclidean Geometry and non-Euclidean — systems where postulate five is changed.

Certainly some, if not all, of these items seem quite basic and self evident. The Parallel postulate may be considered an exception to this and it was a problem for Euclid and many mathematicians that followed. However, as I’ve stated, these are accepted as true without proof. Everything else in Geometry, and all of mathematics for that matter, is based on these basic ideas. They are derived using a formal process called a “proof.” We now understand that all axiomatic systems must start with a few accepted facts that can’t be proven, but must be “accepted.” All else is constructed systematically from these basic beliefs. That is what an axiomatic system is. The constructions must be consistent and follow clear rules of logic. The axioms act as the foundation or basis for it all.

It is absolutely amazing what Euclid was able to do with his basic axioms and the two tools of Geometry. Not only did he derive geometric results, but he explained how to use the tools to produce certain numeric results, such as the square root of two. His collection included books on plane geometry, ratios and proportions, and spatial or solid geometry. Recall that ratios were essential to Greek number theory and the basis of the Rational numbers. Euclid also dealt with Irrational numbers in the construction of roots and even analysis of pi.

One unsolved problem for Euclid was the “squaring of the circle.” That problem is to construct a square with the same area as a given circle using only the two tools of compass and straight edge. We now know that that problem can’t be solved since pi is a transcendental number. Besides being Irrational, it can’t be produced by a finite number of additions, subtractions, multiplications, divisions, powers and roots. That means you can’t solve the problem of the area of a circle with only a compass and straight edge.

As an educator I was always interested in student’s reactions to mathematics. Many would struggle with more advanced arithmetic and algebra, and yet they would come alive and gobble up Geometry lessons like you would not believe. I think the ability to sort of "start fresh" when they first experienced Geometry was part of the effect. After all, High School Algebra is based on Junior High logarithms and Elementary School fractions. Once a student got behind in traditional math, they had trouble catching up because it all builds on previous learning.

Geometry sort of started all over fresh, and allowed those with weak backgrounds in math to begin from the beging. In addition, many students would respond better to the visual representations of Geometry and that helped them understand better than math based on numbers or letters. The focus in Geometry seems different too. Instead of solutions to equations, the problems were often to define proofs, which seem more like legal arguments than mathematics. In any case, people often report that they enjoyed Geometry more than any other math.

As an aside, the ability to do proofs is not good in modern college students. Many universities will teach a refresher course on mathematical proofs for students who pursue advanced math. I believe that, decades ago, students understood proofs better due to more study of Geometry in High School than is done today.

However, while the visual nature of Geometry makes it initially more accessible than other parts of mathematics, such as algebra or number theory, geometric language is also used in contexts far removed from its traditional. Sadly, as I stated, I think modern K-12 school curriculum have reduced the amount of time spent on Geometry in favor of arithmetic and algebraic topics. In my day I studied Geometry for a year and one-half in High School, plus half a year of Trigonometry and two years of Algebra. Now we often teach an introduction to Calculus in High School, but replace Geometry with this advanced form of Algebra.

As I mentioned above, Euclid had concerns about the fifth or “Parallel Postulate.” He didn’t use it in a proof until proposition 29, proving the first 28 without its use. It seemed too long and just not as basic as the other nine. Almost immediately after the publishing of Euclid’s Elements in ancient times, other mathematicians were critical of it. It just didn’t seem to be self evident and too complicated to be basic. However, all attempts to prove it based on the other nine axioms failed.

One of the principles that Euclid and his early contemporaries followed was that this math system they built was intended to represent the real world they lived in. That is what is meant by the axioms and postulates being “self-evident” and "Earth measurement."

As mathematics matured, scientists started to realize that Geometry, and all of mathematics, was primarily a logical system built on basic and simple assumptions. It was no longer considered that important that it matched the physical world. Even though physical sciences would often use math to measure and prove concepts about the natural world, mathematicians were not convinced that math had to even be “practical.” It was a worthy system in and of itself. More than once mathematicians would develop a branch of math for its own beauty and structure, only to learn later it could be applied to certain physical problems.

For at least a thousand years, geometers were troubled by the disparate complexity of the fifth postulate, and believed it could be proved as a theorem from the other nine. Many attempted to find a proof by contradiction, including Persian mathematicians Ibn al-Haytham (11th century), Omar Khayyám (12th century) and Nasīr al-Dīn al-Tūsī (13th century), and the Italian mathematician Giovanni Girolamo Saccheri (18th century). A proof by contradiction starts assuming a fact is false and then goes on to show that would be a contradiction to the existing system. Therefore, it must be true. So these mathematicians would assume that the postulate was false and then try to develop a result (proof) where that was a contradiction. However, that never worked. Seems that mathematics can be developed that assumes the axiom is false. Yet we know from experience that the rails on a railroad (parallel lines) never meet.

Finally, math learned to embrace the contradiction. They realized there is a consistent Geometry where the fifth postulate is false. Bernhard Riemann, in a famous lecture in 1854, founded the field of Riemannian Geometry, discussing in particular the ideas now called manifolds, Riemannian metric, and curvature. He constructed an infinite family of geometries which are not Euclidean by giving a formula for a family of Riemannian metrics on the unit ball in Euclidean space. The simplest of these is called elliptic Geometry and it is considered to be a non-Euclidean Geometry due to its lack of parallel lines.

A simple view of Euclid’s parallel lines axiom is that two parallel lines will never meet. All other straight lines must intersect — once. Non-Euclidean Geometry assumes that parallel lines do meet or non-parallel lines may cross more than once.

However, most scientists still thought the physical world was “Euclidean.” In the early twentieth century, Einstein demonstrated that space is distorted or curved, a phenomenon caused by both mass and energy. Low and behold, the physical world is non Euclidean. I’ve always thought that was a great irony.

To understand this concept of warped space you have to form an analogy from plane or two dimensional space. Consider most maps. They are two dimensional representations of the earth’s surface, Yet we know the earth’s surface is not flat, it is the surface of a sphere. Of course, for small distances it doesn't’ matter that much, but attempting to display the entire surface of the earth on a flat map causes distortion of size. There are many different methods or “projections” that attempt to reduce the error.

Now consider further an flat map of the Earth. We know there are lines running north and south on maps and globes called longitudinal lines. These lines cross the equator at a right angle, so, by the fifth postulate and its implications, they are parallel. Yet they meet at the poles. We understand that because we know that the lines are on a solid figure called a sphere and we expect that behavior. But if we only view the Earth as a flat plane, then that would be “non-Euclidean.”

Now imagine space in four dimensions. It can be curved and act like the globe and “bend” straight lines, at least that is how it appears to us in three-space.

So we understand how a three dimensional world is distorted when represented on a flat surface or map. Now imagine how to think in four dimensions and understand the distortion of three-space. It’s pretty hard to do.

But it is pretty simple with mathematics. Our current understanding of physics requires thinking in more than three dimensions. Einstein added time as a fourth dimension, but it is more complicated than that when you discuss the "curvature" of space.

Although Euclid’s work contained both shapes and numbers, it was primarily a geometric view of numbers. So the division between the two branches of math: geometry and arithmetic/algebra remained for over a thousand years. Eventually, however, the two disciplines were combined and now we often consider geometry and numbers, arithmetic, algebra, etc. as two sides to the same coin.

Next we will talk about a branch of math that, early on, merged shapes and numbers. We call this branch “Trigonometry” and it started with triangles, but quickly became much broader in application.