Paul Erdős (26 March 1913 – 20 September 1996) an outstanding mathematician; also known for his eccentric lifestyle once said:
“Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is”.
One would be tempted to say the same for mathematics but what could be be true for numbers isn’t, according to me, for mathematics. Though they are both beautiful I am convinced that it is easier to show the beauty of mathematics to someone who in first instance failed to see it.
Since years I am working on a book on the Beauty of Mathematics, I don’t know if one day I will finish it but in the meantime I frequently discuss this issue with people often just to test some of my ideas.
Recently I had a very interesting discussion with two students; Chen dingqiang陈定强 (Leo) and Wang Ge王歌 (Alisa) (see also Hangzhou West Lake). Early in the morning we left for a visit of the Lingyin Temple (灵隐寺;靈隱寺; Língyǐn Sì); a Buddhist temple located in the mountains north-west of Hangzhou.
It was a rainy day and after having walked for many hours we decided to have lunch at another Buddhist Temple ; Yongfu (永福寺)famous for its tea fields.
We were sitting on a terrace, sheltered from the rain, just in front of the Teahouse part of the Yongfu Temple. From surrounding nature emerged a romantic mood for reflection. First we went back to a discussion we had the day before on “destiny” and shortly thereafter inspired by the beauty of the landscape I mentioned the beauty of mathematics and in particular one of my favorite formulas.
I wrote down on a slip of paper Euler’s formula eiπ + 1 = 0 who is certainly the most beautiful formula of mathematics.
Alisa looked at it and somewhat to my surprise she asked where the beauty in this formula is? I explained that the beauty lies in the fact that this formula combines in one formula the transcendental numbers: π and e, the imaginary number “i”, and two special numbers: one and zero.
The formula can also be written as: eiπ = -1 meaning that the combination of the two transcendental numbers π and e and the imaginary number i gives (-)1 ! Amazing! (at least for me).
It took, however, some time before the meaning of beauty as I see it in mathematics was understood by Leo and Alisa.
I then asked them which would be their most beautiful formula. Leo wrote immediately down in polar coordinates:
r = 1 –sin(ϴ) .
I recognized the formula but I didn’t remember the details of it.
To show what he meant with beauty he drew the graph of this function.
I was astonished; the beauty he sees in this formula lies in what the graph of this function symbolizes. In working on my book I have listed a number of interpretations of beauty in mathematics and one of them relates to the beauty in graphical representations of formulas as a work of art but I never had considered Leo’s interpretation.
For those who are not familiar with polar coordinates: In two dimensional polar coordinates a point is determined by two coordinates: its radial coordinate or radius, and its angular coordinate or polar angle
Whereas a point in Cartesian coordinates (two dimensions) a point is determined by two coordinates: the value of x : abscissa and the value of y: ordinate.
To relation between the two sets of coordinates is shown on the picture i.e.
y= r . sin(ϴ); x= r . cos(ϴ) and r² = x² + y².
The formula r = 1 – sin(ϴ) transformed into Cartesian coordinates using the above relation becomes x² + y² + y = (x² + y²)1/2. A very complex formula showing the usefulness and beauty of polar coordinates.
Playing with formulas is always fun. Assume now that we do not transform the formula from polar coordinates into Cartesian coordinates but that we just interchange in r = 1 -sin(ϴ) the two polar coordinates by the two Cartesian coordinates i.e. y = 1 – sin(x) what would be the graph of this function?
The result is rather disappointing it is a sinus that has been shifted by π rad on the x-axis and by centered on y = 1 as shown in the upper graph. On the graph below is shown the normal sinus function.
When comparing these two functions at each point along the x axis it is evident that the sum of the two functions is equal to 1.
I was really happy with Leo’s proposal it shows another interpretation of the graphical representation of formulas as beauty in mathematics. In fact, one cannot say that the beauty resides in the graph itself, but what counts here is something that lies beyond the drawing, in what it reminds us of. Looking at it we recognize directly a heart and from there it brings us to our own interpretations of what a heart inspires us of.
Besides this beauty, this formula also illustrates the beauty of polar coordinates.
After Leo wrote down his formula Alisa proposed: 1 + 1 = (?)2 as her favourite formula. I looked at it and thought that she meant that 1 + 1 equals not necessarily 2, it could be also 10 in binary numbers (numbers base 2) but I didn’t ask her for further explanation.
After this break Leo and Alisa proposed to climb a mountain to visit another temple.
When I was writing this article I thought that I should verify with Alisa if she really meant my binary interpretation. A few days later she gave me a total different clarification showing that I was wrong. For more clarity she wrote it in Chinese and translated it would read as follows:
In mathematics, we know at a very young age that 1 + 1 = 2 , but why is 1 + 1 = 2 ? Has it been proven? This problem seems simple but marvelous. It is the foundation of all mathematical theorems, but it cannot directly be proven mathematically. It can only be proven by contradiction: Suppose the 1 + 1 does not equal 2 , then mathematics and mathematics used in all places will be in a mess, so 1 + 1 must be equal to 2 .
The formula may seem simple, but it is about a “number” on the basis of a formula, without it, we would not have mathematics, physics and other natural sciences. This formula has a special meaning for human beings to know the world.
In mathematics, this formula has been established as the fundamental theorem, but there is no limitation of the scope of mathematics , it may not be necessarily true , the most simple example: a drop plus a drop of water is still a drop of water . Therefore I express it as 1 + 1 = ( ? ) 2. This formula is very interesting and the beauty of this equation resides in the reasoning underlying it.
Alisa’s contribution is very interesting since it illustrates another interpretation of “Beauty” in Mathematics.