Irrational numbers are values that can not be expressed with a finite decimal expansion. They show no pattern as you expand them to larger and larger number of digits. Pi is the ratio of a the circumference of a circle to its diameter. Pi to 50 digits is 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510, but that still isn’t exactly the value of pi.
“e,” the base of natural logarithms, is is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series. It is sometimes called Euler's number after the Swiss mathematician Leonhard Euler. e to 30 digits is 2.7182818284 5904523536 0287471352 6624977572 4709369995 and it keeps on going too.
And imaginary numbers are based on the square root of minus one. That’s a number that can’t exist and can only be imagined. Yet it is surprisingly useful in math and physics to describe the real world.
The fact that all these strange and some “unending” numbers can be combined to make the simplest number there is seems like the punch line of some celestial joke that we just don’t get.
(See my earlier note on what many feel is the “The Most Beautiful Formula in All of Mathematical Thought.”)
There are many other equations, conjectures, and formulas that mathematicians find most significant for their aesthetic charms. The symmetry and mystery of math is as much a cause for our wonder as is our artistic appreciation of a natural object such as a beautiful sunset, a flower, or the view of a powerful mountain range.
On the other hand, you often find mathematics in beauty. The “Golden Rectangle,” (and the underlying golden ratio and Fibonacci Series) a pure mathematical concept, is found frequently in drawings, paintings, and architecture. It is often the basis of classical views of beauty and art, even by those that didn’t recognize its mathematical significance.
In fact, the specific mathematical object that is the subject of this essay is related to the Golden Rectangle.
Of course the motivation and realization of both mathematics and beauty is often found in nature. This is particularly true of my discussion today of the Spira Mirabilis.
Of the numerous mathematical curves we encounter in geometry, in nature, and in art, perhaps none can match the exquisite elegance of the logarithmic spiral. This famous curve appears, with remarkable precision, in the shape of a nautilus shell, in the horns of an antelope, and in the seed arrangements of a sunflower.
It is also the ornamental motif of countless artistic designs, from antiquity to modern times. It was a favorite curve of my favorite artist, the Dutch illustrator M. C. Escher (1898–1972), who used it in some of his most beautiful works, such as Path of Life II.
From a mathematical perspective, The logarithmic spiral is best described by its polar equation, written in the form r = eaθ, where r is the distance from the spiral’s center O (the “pole”) to any point P on the curve, θ is the angle between line OP and the x- axis, a is a constant that determines the spiral’s rate of growth, and e is the base of natural logarithms. Put differently, if we increase θ arithmetically (that is, in equal amounts), r will increase geometrically (in a constant ratio).
The many intriguing aspects of the logarithmic spiral all derive from this single feature. For example, a straight line from the pole O to any point on the spiral intercepts it at a constant angle α. For this reason, the curve is also known as an equiangular spiral.
As a consequence, any sector with given angular width Δθ is similar to any other sector with the same angular width, regardless of how large or small it is.
This property is manifested beautifully in the nautilus shell. The snail residing inside the shell gradually relocates from one chamber to the next, slightly larger chamber, yet all chambers are exactly similar to one another: A single blueprint serves them all.
The logarithmic spiral has been known since ancient times, but it was the Swiss mathematician Jakob Bernoulli (1654–1705) who discovered most of its properties. Bernoulli was the senior member of an eminent dynasty of mathematicians hailing from the town of Basel. He was so enamored with the logarithmic spiral that he dubbed it “Spira Mirabilis” or Spiral Wonderful, and ordered it to be engraved on his tombstone after his death.
His wish was fulfilled, though not quite as he had intended: For some reason, the mason engraved a linear spiral instead of a logarithmic one. (In a linear spiral the distance from the center increases arithmetically — that is, in equal amounts — as in the grooves of a vinyl record.)
The linear spiral on Bernoulli’s headstone can still be seen at the cloisters of the Basel Münster, perched high on a steep hill overlooking the Rhine River.
But if a wrong spiral was engraved on Bernoulli’s tombstone, at least the inscription around it holds true: Eadem mutata resurgo— “Though changed, I shall arise the same.” The verse summarizes the many features of this unique curve.
Stretch it, rotate it, or invert it, it always stays the same. BEAUTIFUL!