Table Proofs

Introduction

I'm always on the lookout for ways of visualizing mathematical results. This might mean drawing a picture, drawing a sequence of pictures, building physical models and playing with them, or even constructing a virtual 3-D object and playing with it via keyboard and mouse.

In this essay I'm going to restrict myself to pictures that clarify a result: very much like the "Proofs without words" articles that the MAA's Mathematics Magazine has been publishing for many years. Furthermore, I'm going to restrict myself to pictures that arise from html tables and can be displayed by internet browsers: plain old fashioned static html with no javascript or images. I like to think of it as an html analogue of ascii art: maybe it should be called "table art".

This means that I'm going to have to leave out a lot of really neat pictures. For example, there's a dissection of an n by n+1 by 2n+1 brick into two almost congruent parts (they have different handedness), each of which can be dissected into three congruent parts, and each of those is clearly of volume 1²+2²+…+n². You're not going to see the picture for that in this essay! But, even though we're restricted to using simple pictures consisting of rectangles with only horizontal and vertical sides, there's still quite a few results that can be visualized.

I should mention that this essay was motivated by the Fibonacci via Pascal result mentioned in this essay. I was trying to think of how to use an html table to illustrate the result, and suddenly realized I could illustrate a lot of other results also.

Arithmetic progression

Gauss derived the formula for the sum of an arithmetic progression when he was 5 years old. It's not known how he did it. Maybe he followed the Feynman technique: write down the problem, then think really hard, then write down the answer. Who knows! I like to believe that he visualized it like this

first         last

= 1/2

first last                 last first

Sum = n(first+last)/2 = n * (average of first and last)

where n is the number of elements.

An infinite geometric series

When we were both teenagers, fellow mathematics enthusiast Peter Williams drew me a diagram very much like the following (it was more of a pie chart, but the idea's the same). Suddenly the convergence of geometric series made sense!

1	2	3	1
			2
			3
			1	2	3	1
						2
						3

1/3 = 1/4 + (1/4)² + (1/4)³ + (1/4)⁴ + …

There's nothing magic about 3 or 4. The same kind of diagram shows that

1/k = 1/(k+1) + 1/(k+1)² + 1/(k+1)³ + 1/(k+1)⁴ + …

or, if you prefer

1/(n-1) = 1/n + 1/n² + 1/n³ + 1/n⁴ + …

Sum of squares of Fibonacci numbers

I know of no better way to understand the following result

12	12	32	82
22
52

1²+1²+2²+3²+…+F_n² = F_nF_n+1

Incidentally, if you think of building up to the final rectangle by continuing to add larger and larger squares, there's a choice at each step: do you place the new square on one side or the other. If you want, you can draw the squares so that they spiral around the initial seed. Here I've chosen to always add to the right, or down, depending on the step.

Fibonacci via Pascal

After reading my essay on Fibonacci numbers, David Angell sent me an email telling me about a result relating the Fibonacci sequence with the binomial coefficients. To quote David: This gives the Fibonacci numbers as sums of "diagonals" in Pascal's triangle. … just drawing up the triangle and colouring in two consecutive diagonals makes it beautifully obvious how they fit together and satisfy the Fibonacci recurrence.

											1
										1		1
									1		2		1
								1		3		3		1
							1		4		6		4		1
						1		5		10		10		5		1
					1		6		15		20		15		6		1
				1		7		21		35		35		21		7		1
			1		8		28		56		70		56		28		8		1
		1		9		36		84		126		126		84		36		9		1

Fn+1	=	(	n	)	+	(	n-1	)	+	(	n-2	)	+	…	=	Σ		(	n-k	)
			0				1				2						{k ≥ 0 and 2k ≤ n}		k

Sum from 1 to n

Following the diagram for an arithmetic progression, we have

= 1/2

(1+2+…+n) = n(n+1)/2

Another possibility is to slightly separate the two copies

2 * (1+2+…+n) = (n+1)² - (n+1)

Yet another possibility is to over-push the two copies together, so that they share the diagonal

2 * (1+2+…+n) = n² + n

Sum of odd numbers

Reading down the page, from left to right

1
	3
		5
			7
				9

1+3+5+…+(2n-1) = n²

Or, if you prefer:

2n-1
	2n-3
		…
			3
				1

n² = (2n-1)+(2n-3)+..3+1

Odd squares

(2n+1)² = 4(2n) + 4(2n-2) + … + 4*4 + 4*2 + 1

As an aside, this makes it clear that odd squares must be of the form 8n+1. This, together with the fact that even squares are of the form 8n or 8n+4, shows that three squares can never have a sum of the form 8n+7.

Even squares

(2n)² = 4(2n-1) + 4(2n-3) + … + 4*3 + 4*1

Incidentally, dividing both sides by 4, we have yet another way of seeing that

n² = (2n-1) + (2n-3) + … + 3 + 1

Sum of squares of Generalized Fibonacci numbers

G0G1	G22	G42
G12
G32

G₁²+G₂²+G₃²+G₄²+…+G_n² = G_nG_n+1 - G₀G₁

Zeno's paradox

…

1 = 1/2 + (1/2)² + (1/2)³ + (1/2)⁴ + …

An infinite sum via quadrants

				…

1 = 3(1/4) + 3(1/4)² + 3(1/4)³ + 3(1/4)⁴ + …

Concluding words

I expect that there are many more mathematical results that can be illustrated by appropriate use of html tables. Feel free to let me know about them.

Feedback

If you have corrections, additions, modifications, etc please let me know mailto:walterv@gbbservices.com