Why do all shapes lie in the Polyhedron Plane?

I'll be doing An Evening of Unnecessary Detail at the Bristol Beacon, UK on Tuesday 12 March 2024. Come and celebrate Almost Pi Day with me! And celebrate Ages Until Tau Day with Steve Mould.

CORRECTIONS
- None yet, let me know if you spot anything!

Filming by Matt Parker
Editing by Alex Genn-Bash
Written and performed by Matt Parker
GeoGebra graphics and addition material by Ben Sparks
Edge complaining by Grant Sanderson
Yes they bent the rules by Henry Segerman
Produced by Nicole Jacobus
Music by Howard Carter
Design by Simon Wright and Adam Robinson

MATT PARKER: Stand-up Mathematician

[CivilWarWeekByWeek]

I have a correction at this point Ben Sparks isn't kindly making geogebra programs he is clearly in your basement forced to code against his will

[johnchessant]

16:42 just to spell out what Grant is saying here, if you calculate V - E + F for a polyhedron that has a hole in it (e.g. if you approximated the surface of a torus with plane faces), then you won't get 2. Instead, you'll get 2 - 2g, where g is the number of holes. So this is a way to formalize the notion of "holes" (since you can just count them via vertices, edges, faces) and prove that the number of holes is invariant with respect to continuous deformations.

[prdoyle]

I love that this video includes Grant being extra Grant and Henry being extra Henry.

[MadSpacePig]

6:30 Funny that you demonstrated a simulation of the polyhedra being projected onto a plane, when in fact, due to the nature of them being rendered on a computer, and displayed on a flat screen, they were already being projected onto a plane, just by us looking at them.

[internetuser]

The United States education system uses "y = mx + b" for the equation of lines.

Also, big fan of the "technically correct if you're a topologist" entries.

[coltonchinn]

7:13 TIL that “way way way more faces” is equivalent to “two more faces”

[HunterJE]

I feel like the easiest shortcut to understanding the "why" of the symmetry of duals is that a dual is very much by definition what you get if you swap the things being counted by two of our three variables for one another (while keeping the thing counted by the third constant...)

[reddcube]

The dual line is easy to explain. One shape and its dual are reflections of each other along the line. That is because when making a dual shape, each Vertex becomes a Face, each Face becomes a Vertex, and each Edges just changes orientation. So reflections of the line is just swapping the V and F.

[JohnJohn]

1:40 I mean.. 3 Blue 1 Brown was just sitting right there...

[japanada]

Why you get lines and not just planes:

For any polyhedron with only triangular faces, you have the additional relation 3F=2E (each face touches three edges, and each edge touches two faces). The intersection of V-E+F=2 and 3F=2E gives a line that contains all polyhedra with triangular faces.

It just so happened that the only polyhedra Matt used in his visualization were triangle-faced polyhedra and their duals (which satisfy 3V=2E, giving the other line). There are lots of polyhedra that don't lie on either line that just didn't get drawn - but the triangle-faced ones and their duals are definitely quite common! (In particular, every platonic solid or its dual is triangle-faced)

[nathanielpranger]

From my experience in the Netherlands we use "y = ax + b". Nice and clear that we use the first two available letters for unknown parameters, so I thought everyone did. Then I saw you use "m" and I just felt sorry for 14-year-olds learning Newton for the first time.

[zahirgizzi]

about 12:58:
I studied in Germany (Leipzig to be precise) and we learned is as y = mx+n 😆

[JamesWanders]

Bad news, Matt. When you said we should go "marvel" at the display, the auto-caption wrote it as "Marvel" so your channel belongs to Disney now.

[GeekRedux]

Given how prevalent TI-80-something graphing calculators are in the US, I'm surprised we haven't seen a shift from
y = mx + b to y = ax + b, since that's how those calculators have always presented it.

[JohnDoe-tinp]

In his memoir, mathematician Goro Shimura says that he once set an exam question for a student who was trying to transfer from another university, which went something like this: Find the equation of the line in the plane that passes through the points (1, 5) and (1, 2). He wanted to see if the student would blindly use the formula y = mx + c. The student fell into the trap and then complained about being tricked.

[sachacendra]

Here in Switzerland we used a multiplicity of letters for the line: ax+by+c=0 or y=ax+b or y=px+q or y=mx+h or y=px+h were all things i encountered in my education. I believe the goal was to teach us that the letters didn't really matter. Also, since Swiss education is very decentralised and each teacher can more or less choose the material they want to use i wouldn't be surprised if elsewhere in Switzerland they would use completely different letters.

[zoerycroft]

im so proud of myself, i knew nothing about this before the video, never even thought about arranging any polyhedra or anything, and when you were saying "well, , what different ways can we arrange them" i said... "i bet the euler characteristic is what makes it a plane"

[georgebayliss]

England (UK), GCSE: y = mx+c

A-level: Very rarely told to give in the y = mx +c format, most commonly we leave in the format y-y1 = m(x-x1) or ax+by+c

[crowman]

Very interesting way of interpreting and visualising Euler's polyhedron Formula

[KerryHallPhD]

I love the pivot at 12:56 from dismissive frustration to a positive query :D Excellent video all around!