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Big and small shapes! (Patreon)

Published:
2021-06-21 15:02:04
Edited:
2021-06-21 17:15:26
Imported:
2021-06

Content

Ok, I have two ideas to bounce off you all which have nothing in common other than they are both kind of "Is shape superlative?" questions. And these are certainly not videos I'll be able to make for a while.

1. Is the ancient city of Angkor Thom the world's biggest square? The outer edge is an exact square with sides of 3km. I've been meaning to do a video about how Washington DC was going to be an exact square with 10 mile edges until Virginia took their bit back. But Angkor Thom is an actual 'solid' square (with a proper edge; it's not just a marked border). Let me know if you can think of another contender!

2. Is the phytoplankton Braarudosphaera bigelowii the world's smallest regular dodecahedron? I know there are virus particles like Adenovirus Dodecahedron but is that really a dodecahedron or is it just a clump of molecules with dodecahedral symmetry? And come on, look at the Braarudosphaera bigelowii, it's so incredibly a dodecahedron.

3. Let me know about any other shape suggestions (other than pointing out that CGP beat me to making a video about the hexagon on Saturn).

Comments

Cain

That is extremely a dodecahedron! I agree with you that viruses do not count as 'biological'. Quasi-biological, maybe.

Hari Krishnan

I'll be more interested in topic 2. Phytoplanktons don't learn math( as far as I know), and thus it'll be interesting to see where the shape comes from. Bees use "best-a-gons" for packing efficiency.(sorry for doing the thing you asked me not to do). Is there a similar geometric reason here?

Laura

I have a strange thought: is the structured layout of DC more conducive to “fitting in the most buildings” versus older winding layouts like Paris? I (think I) know that DC was designed purposefully so that it was easily navigable, which it generally is (except for all the one-way restrictions). But is there any other ‘geometry of a modern city’ benefit to that kind of structure?

Clinton

There's an artificial compound called Dodecahedrane - https://en.wikipedia.org/wiki/Dodecahedrane - which is probably the world's (possibly universe's since we created it?) smallest regular dodecahedron. It's not nearly as pleasing to look at as the Braarudosphaera bigelowii though!

Frank Gevaerts

Never mind smallest... An actual living pefect dodecahedron surely proves that mathematics is the foundation of this universe!

Matt Parker

On one hand, it is just 20 carbons-hydrogen pairs arranged like a dodecahedron, whereas Braarudosphaera bigelowii looks so incredibly dodecahedrony. But on the other hand "Smallest regular dodecahedron in the universe" is pretty cool. I can overlook a lot of aesthetics for that.

Matt Parker

They come from a family which build sphere-ish shapes out of calcified scales, and pentagons are the best-a-gons (sigh) of spheres.

Matt Parker

I've actually not looked into the history of why they picked such a perfect square (albeit on a jaunty angle).

Matt Parker

Yeah, clumps of atoms pretending to be alive. Honestly.

Nick Poulos

It sounds like you're looking for large, marked squares, but as for "unmarked" political divisions, the US has plenty of larger ones, at least to the extent of 1800s surveying skill. Much of the western United States is divided into 6 mile square townships. In some parts of some states, these creations for surveying are used for governmental boundaries themselves. Often, they were aggregated and formed the building blocks for counties as well. So if you look at Miner County, South Dakota, it is pretty much a 24 mile square—16 townships—as are a bunch of counties in southern Nebraska, for instance. There are plenty of squarish bigger examples, but surveying errors kind of make them a bit wonky. I haven't gone and measured, but Cochise County, Arizona, Custer County, Nebraska, and Harding County, South Dakota look suspiciously squarish. And there's also the White Earth Reservation in Minnesota—Google Maps has a chunk taken out of it, but based on the US BIA and the tribe themselves (https://whiteearth.com/assets/img/community/White%20Earth%20Base%20Map%2024x36%20w_Allotments.pdf) it's a 36-mile square.

ViliamF

A little biology here. Viruses are non-cell organisms, thus their entire body is really small, and composed of molecules. E.g. those thorns on COVID viruses are proteins, which help the virus get into the cell - kind of like a lot of fake IDs. The dodecahedron virus is thus a true dodecahedron down to the precision of atoms.

Nick Poulos

Essentially, two-thirds of the US is squares. I could go on and on about the Public Land Survey System (PLSS) but I'll leave you with one more thing: they laid this grid on the US without knowing anything about the underlying topography! Fine for Kansas, not so great for Colorado.

Jeff Raymakers

No discussion about surprising naturally occurring shapes would be complete without buckminsterfullerene (aka buckyballs). Sixty carbon atoms arranged in a precise truncated icosahedron (e.g. the football/soccer ball shape). I know you like your (geometrically correct) football/soccer balls.

Jeff Raymakers

For large squares, you might consider the LIGO experiments. While they don't delineate all four sides of a square, they do mark out two 4km lengths at right angles. And they do this extremely precisely. And there are two of them.

Bryn

I prefer topic number 1.

Arnim Sommer

Don't forget Buckminsterfullerenes, and didn't some greek philosopher corellate the then known planets with the platonic solids?

Matt Parker

Oh yes, I wrote "cells" when I meant "molecules". I'll edit that in the post to avoid further confusion. And I take your argument for that virus being a true dodecahedron onboard.

Matt Parker

I think Kepler had some platonic solid thoughts about planet orbit ratios.

David Lake

Sorry - I can't stop glitching on the word 'exact'. How precise is that? How accurate? (I'm not even going to mention the 0.71m curvature of the earth because that would make me an even more sad old pedant.)

Mik Scheper

It's the first thing I thought about, too. The planet not being a perfect sphere doesn't help, either.

PeteyPak

Haha! Trick question! It's you Matt, you teach mathematics and puzzle solving on the internet, you're not just square, you're the world's biggest square.

Mik Scheper

If rectangles count, I've always wondered about Wyoming and Colorado, whether they have any anomalies, either in the way they're defined or in practice on the ground. But then we get into the whole flat earth problem anyhow.

Mik Scheper

If we ignore the spherical earth problem and allowed freedom of protection, I'd say the southern hemisphere would be the biggest square.

Paddy Kenworthy

And we pay money for the pleasure. Maybe collectively we’re the biggest squares? 🤔

Henry Segerman

What is the difference between a dodecahedron and a clump of molecules with dodecahedral symmetry? You're not going to find any actual straight lines in real life to make the edges of a dodecahedron, everything is an approximation! Vi Hart and I discuss a related question on page 2 of http://archive.bridgesmathart.org/2014/bridges2014-143.pdf

The Live Toad

If you do squares I'd like to know whether being closer to the equator gives Angkor Thom an advantage anyway -- it's one thing to make a geodesic quadrilateral with 4 right angles and another to make a square. How do DC and AT do with respect to both of these measures?

Greg Jensen

Surely, by combinatorial enormity, there is some combination of English ley lines that forms a square to within a particular tolerance.

Greg Jensen

Additionally, are we talking about squares within Euclidean or Spherical geometry? On the scale of 3km, I can imagine there would be small but measurable differences in the curvature of the edges, even if the angles themselves are still too close to 90° to tell the difference.

Stuart Ward

This is the proof that the earth is not flat, that a planar object has angles greater than 90 for a square. On the size of a 3km side I am sure that modern survey equipment would be able to measure this.

sim1

The Very Large Array is 20+ miles across and I assume laid out very carefully. Makes it bigger that 10mile square.

Matt Parker

I think I'm going to define "dodecahedron" as "really looks like a dodecahedron".Or maybe in the case of the virus particle: doesn't have disproportionate sticking-out bits".

Matt Parker

I looked into the exact shape of Colorado a while back and it's actually got a bunch of tiny angles in it. Plus: with the borders between states I assume there is no wall or line to actually mark them out.

Matt Parker

and in terms of the iceberg: my wife Lucie suggested that on some planet somewhere a square will have formed along those lines. So I reframed the question to be "in the world". Problem solved.

Matt Parker

Nice suggestion! But I doubt they'll move them together for me.

Matt Parker

I am on the case the find some. Doesn't seem likely though.

Matt Parker

Doesn't look like anyone has made any yet (and that paper suggesting it was 2014).

Matt Parker

But do they have a square fence around it?

PeteyPak

I like your creative use of constraints. But those are Earth icebergs! It's a NASA mission that's looking back down at us.

PeteyPak

They _are_ in the world. Or at least they were...

sim1

I guess not, and it has to be a square?

Phil Brady

Lots of roads in the Midwest of the US are layed out in a mile-spaced grid pattern, thanks to an 1789 act iirc. Would that qualify for your square definition?