Some Visual Proofs by Desmos

Here are a number of visual proofs of classic mathematical results I (re)created in Desmos. Most of these illustrations are interactive, so feel free to tinker with the sliders, toggle the folders (by clicking the circle to their left) to enable/disable features, zoom in/out on the right panel, and such.

The demonstrations are loosely grouped in the following categories:

• Sum of Finitely Many Quantities
• Sum of Infinitely Many Quantities
• Geometry and Trigonometry
• Others

Links to these demonstrations are also provided below -- please feel free to adapt them for your own use. If you have any questions/comments/bug reports about these demonstrations, or would simply like to let me know that you find them useful, please feel free to e-mail me at Pau@mathQ.usask.caR (with P,Q, and R removed).

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Sum of Finitely Many Quantities

• The $n$-th triangular number (i.e., $1+2+\cdots+n$) is equal to $\frac{n(n+1)}{2}$.

(Link to Demonstration)

• A positive integer is an odd square if and only if it is eight times a triangular number plus one.

(Link to Demonstration)

• For every integer $n \geq 1$, $1^2+2^2+ \cdots + n^2 = \frac{1}{3} \cdot (2n+1) \cdot \frac{n(n+1)}{2}$.

(Link to Demonstration)

• For every integer $n \geq 1$, $1^3+2^3+ \cdots + n^3 = (1+2+ \cdots +n)^2$.

(Link to Demonstration)

• An alternative proof to the previous result. This one does so by showing that $4(1^3+2^3+ \cdots + n^3) = (n(n+1))^2$.

(Link to Demonstration)

• Let $F_n$ be the $n$-th Fibonacci number for every $n \geq 0$. (So $F_0=1, F_1=1$, and $F_n = F_{n-1}+F_{n-2}$ for every $n \geq 2$.) Then $F_0^2+F_1^2+ \cdots + F_n^2 = F_n \cdot F_{n+1}$ for every $n \geq 0$.

(Link to Demonstration)

Sum of Infinitely Many Quantities

• The series $\displaystyle\sum_{i \geq 1} \frac{1}{i}$ diverges.

(Link to Demonstration)

(One can also see from the above demonstration that $\displaystyle\lim_{n\to\infty} \sum_{i=2^n}^{2^{n+1}-1} \frac{1}{i} = \ln(2)$.)

• The series $\displaystyle\sum_{i \geq 1} \frac{1}{i^2}$ converges to less than $2$.

(Link to Demonstration)

(More generally, it shows that $\displaystyle\sum_{i \geq 2^n} \frac{1}{i^2} < 2^{1-n}$ for every integer $n\geq0$.)

• For every real number $p$ where $-1 < p < 1$, the series $\displaystyle\sum_{i \geq 0} p^i$ converges to $\frac{1}{1-p}$.

(Link to Demonstration)

Geometry and Trigonometry

• (Pythagorean Theorem) Given a right triangle with legs of lengths $a,b$ and hypotenuse of length $c$, the side lengths must satisfy the equation $a^2+b^2=c^2$.

(Link to Demonstration)

• Another (in my opinion, very neat) proof of the Pythagorean Theorem.

(Link to Demonstration)

• The circular disk of radius $r$ has area $\pi r^2$.

(Link to Demonstration)

(Note that the proof only makes use of the definition of $\pi$ being the ratio between a circle's circumference to its diameter, and not our knowledge of the numerical value of $\pi$.)

• The sum-of-angles identities for cosine and sine.

  (Link to Demonstration)

• $\arctan(1/3) + \arctan(1/2) = \arctan(1)$.

(Link to Demonstration)

(In fact, one can tinker with the value of $c$ and the location of the point $P$ in the demonstration to generate other similar $\arctan$ identities.)

• (Viviani's Theorem) Let $P$ be any point inside an equilateral triangle. Then the sum of the lengths of the three perpendiculars from $P$ is equal to the height of the equilateral triangle.

(Link to Demonstration)

Others

• There is a bijection between the open interval $(0,1)$ and the set of real numbers $\mathbb{R}$.

(Link to Demonstration

(In fact, the function shown above is $f :(0,1) \to \mathbb{R}$ where $f(x) = \tan( \pi x - \frac{\pi}{2})$.)

• If a natural number $p$ is not a perfect square, then $\sqrt{p}$ is irrational.

(Link to Demonstration)

(This one requires some additional explanation. The illustration above shows that if there was a positive integer $k$ where $\sqrt{p} \cdot k$ is an integer, then there would be a smaller positive integer $\ell$ where $\sqrt{p} \cdot \ell$ is also an integer, contradicting the existence of $k$.)

• One can rotate $p$ disks inside a larger circle, and place $q$ points on each disk, such that the $pq$ points collectively trace out a $(p+q)$-point star.

(Link to Demonstration)

(Note: The statement above is pretty imprecise! For some values of $p$ and $q$ the points in fact lie outside the rotating disks, and the "star" may or may not look like a star. This is created in response to this video by Mathologer.)

• Finally, as a cautionary tale for relying on visual proofs without thinking critically, here is a very standard proof showing that $25=24$.

(Link to Demonstration)

Like what you saw? You can find more visual proofs by doing an internet search for "Proofs without words". In particular, check out the excellent Mathologer Youtube channel, as well as this post on Mathoverflow, which are the sources of inspiration for some of the visual proofs above.

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