My mental state remained good today, but I also slept a lot. I did some housekeeping on the blog to make it easier for me to return to questions I’ve raised but not answered. My study time was devoted to work on the proposition about inscribed angles subtended by the same arc and to considering whether and how my other recent proofs depend on the parallel postulate. The proof of the inscribed triangle proposition does depend on it, I believe. It is not necessary to the proof that the sum of the angles of a triangle is $\pi$ radians, but it is necessary that it is the same for all triangles, which is not the case in some systems of geometry.
Category: Math Posts
Before and After
Today was devoted to fun with geometry (as was some time late last night). I came up with a simpler proof of the proposition about the inscribed equilateral triangle, which I’m pleased about; the first one was a bit gnarly. I also worked on the lemmas to accompany it. It needs most of the same ones, but not the one about parallelograms.
Here are before and after shots showing the working figure for my first proof and that for my second. One proof required four new lines and two new named points. The other involved just one one of each.
(I’m looking forward to finishing this notebook so that I can move to an unruled one.)
Groundwork
I spent quite a bit of time today working on the inscribed equilateral triangle problem. I experimented both by hand and using the Desmos geometry tool, which allows you to vary parameters in your drawings. At this point, I’ve come up with a proof that convinces me. There is one step that needs to be formalized and a couple that depend on facts that I know to be true but would like to prove. These include that the sides of a parallelogram are pairwise equal1 and that a triangle is equilateral if and only if all of its angles are equal. The argument also depends heavily on a fact I just learned today, conjecturing it based on experimentation and then confirming it with a Google search: inscribed angles subtended by the same arc of a circle are equal (as shown below). I’m not sure whether I will be able to prove that or not. My education in geometry focused almost exclusively on lines and triangles, and I know little of circle geometry.
(As readers may have guessed, I’m feeling much better today.)
- To do. ↩︎
Triangle, Triangle
Today I worked for as long as I could on the proposition from yesterday about the inscribed equilateral triangle. I’m feeling a little better, but my mind is still sluggish, so I didn’t make much progress.
I’m wondering whether proving the proposition for a special case first would be helpful. The edge cases where $D$ is the same point as $A$ or $C$ are trivial. The case where $AD = CD$ might provide some insight. On the other hand, using any of the unique characteristics of that case to prove it could leave me as far from a general proof as before.
Ptolemy’s Theorem and Others
Things continued difficult today, but I felt a bit better after dinner. I watched four videos from the Numberphile channel, notably one about proving a geometric relationship called Ptolemy’s Theorem by a technique called inversion of the plane. It was a great video, but quite involved. I would not recommend it to those without a strong interest in math.
When I am feeling better, I would like to have a go at proving Ptolemy’s Theorem by another method.1 The lecturer in the video said that it can be done using plane geometry, provided you are clever, and can also be done by crunching trig ratios, something I have some experience in. I’d also like to prove, by some other method, a fact that can be proven using Ptolemy’s Theorem: that in the figure shown below, the distance from $D$ to $B$ is the sum of the distances from $D$ to $A$ and $C$.
A cool thing about Ptolemy’s Theorem is that the Pythagorean Theorem falls right out of it. In fact, you could say that the Pythagorean Theorem is a special case of Ptolemy’s. This means that Ptolemy’s Theorem must only be true if you assume the Parallel Postulate. I wonder if that is the case for the proposition above.
- To do. ↩︎
Tilings Again
Today I watched an enjoyable interview with one of the researchers involved in the discovery of the first aperiodic monotile, as described in the second video from my post last Wednesday. (The channel it comes from, Numberphile, has a wealth of other interesting math videos, as well, most featuring mathematical guests explaining their interests.)
Fun with Tilings
Today I’ve been playing with tilings to try to understand the relationship between radial symmetry and periodicity. I’ve come up with the following to show that each can exist independently of the other.
I’m still wondering, though, if a tiling can be radially symmetrical about more than one point and be non-periodic. I suspect not.1
- To do. ↩︎
Shapes on a Plane
Today I watched three videos about tilings, specifically non-periodic tilings. These are arrangements of tiles that cover an infinite plane in a pattern that never repeats.
Two of the videos were quite accessible. The first was from the popular science channel Veritasium1 and covered the state of this subject three years ago:
The second was from a smaller channel called Up and Atom and explained recent developments that have caused excitement among math enthusiasts.
The third video was more technical and probably of less interest to nonspecialist readers. It included a little more detail about why some collections of shapes form non-periodic tilings.
I’m still very tired from yesterday and don’t have much bright to say about all this. I feel like there is something to understand, though, about the relationship between rotational symmetry and periodicity of tilings. That is, between the ability of a tile pattern to match its original configuration when rotated versus when translated.
- Note that Veritasium has sometimes shaped its educational content to reflect well on its sponsors. ↩︎
Discontinuous
Things are a bit disrupted today. I am out of one of my medications and am feeling the withdrawal symptoms more and more. Nevertheless, I read more of Infinite Powers. The section I read contained several interesting ideas. One was that, while calculus is very useful in modeling the natural world, the assumptions that space and time are continuous that are fundamental to calculus may not be accurate. At the smallest scales, space and time may be broken into indivisible segments that span units called Planck length and Planck time. I will have to think more about this and how it relates to the idea that mathematics is inherent in nature.
Proofs That Go Poof
Today I read another section of Analysis with an Introduction to Proof. This one was dedicated to proof techniques, and one of those made use of the tautology $[p\implies (q\lor r)]\equiv [(p\land \lnot q)\implies r]$. This is intuitive enough, but proofs that use it feel like cheating. For instance, to prove that if $x^2=2x$, then $x=0$ or $x=2$, you can assume that $x^2=2x$ and $x\neq0$, then prove that $x=2$. That’s it. Poof! Done! No showing that $0^2=2(0)$ required. I do not find this intuitive.