@henryseg@mathstodon.xyz
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henryseg

@henryseg@mathstodon.xyz

Mathematician working mostly in three-dimensional geometry and topology, and mathematical maker/artist working mostly in 3D printing and virtual reality.
#Math #Maths #Art #3DPrinting #Geometry

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Danpiker, to random
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video/mp4

henryseg,
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@Danpiker I don’t know what’s going on but it’s cool.

henryseg, to random
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2023 conference spherical photo.

gregeganSF, to random
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A sphere is the nicest surface with constant Gaussian curvature κ=1, but these surfaces also have κ=1, except at the cusps.

In cylindrical coords:

z = √(1-χ²) E(arcsin(ρ/χ) | χ²/(χ²-1))

z is measured from a cusp
χ is radius of equator
E is an elliptic integral of the 2nd kind

A sphere deforms, stretching along one axis and shrinking in width into something like a prolate ellipsoid, but with a cusp at the poles.

henryseg,
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@ngons @gregeganSF Is it cool to do impossible things? Doing nearly impossible things is often cool, but I’m not sure that property extends to the limit.

benleis, to random
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Infinity decahedron we found at the Seattle art fair.

henryseg,
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@benleis Do they call it a dodecahedron?

henryseg,
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@benleis I think you may be mixing up the dodecahedron and the icosahedron. The shape in the photo looks like it is made of triangles with five at each vertex, which would be an icosahedron?

henryseg, to random
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Only took 14 years from my first YouTube video!

adereth, to random

My poor family tried to get me a nice gift, but I informed them a coffee cup is already a donut and this is just some freaky genus-2 torus.

A Stan’s Donuts coffee mug featuring a hole going through the body of the mug, viewers from the top.

henryseg,
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@hungryjoe @adereth It depends on interpretation. You could ask what the boundary surface of the mug is, or you could also ask about the surface you get by ignoring the thickness of the mug. For example, you can think of a dinner plate as a sphere if you care about the boundary, or a disk if you ignore the thickness.

henryseg,
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@hungryjoe @adereth The boundary of your mug looks like a genus three surface, while the version ignoring thickness looks like a torus with a boundary component, and (if the handle has a bit of a flattened cross-section) with a rectangular strip glued onto it at either end of the rectangle.

henryseg, to random
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Test shot for a 3D printed fractal tree.

henryseg,
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@eric Tastes like photo sensitive resin...

richardelwes, to random
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Four beautiful dice in the shape of Catalan solids, AKA Archimedean duals. These shapes have identical faces (perfect for fair dice), but irregular faces (unlike Platonic solids, e.g a cube). The 12 & 30 sided dice here have rhombus faces, and the 24 & 60 ones have kites.

Link to the Catalan solids on Wikipedia - they’re some of the most beautiful polyhedra around in my opinion.

https://en.m.wikipedia.org/wiki/Catalan_solid

henryseg,
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@richardelwes The d30 is a countdown (also called spindown) die which we don't make, and the numbering on the rhombic d12 is different from ours. But the d24 and d60 look like they were made by us (or were made by someone who copied our numbering): http://thedicelab.com

henryseg,
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@wrog @richardelwes We did a 120-sided one! https://youtu.be/516U4whg4GU
To add to the Catalan dice list, we also have a disdyakis dodecahedron d48 and our "dLX" 60-sided alphabet dice are pentakis dodecahedra.

henryseg,
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@wrog @richardelwes The problem with them is that they don’t have parallel faces on opposite sides. So when the die is lying on a face, it isn’t clear which face is on top. So how do we write the numbers?

henryseg,
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@richardelwes @wrog Yes, and with 60 sides, the numbers have to be quite small… By the way, that d7 doesn’t look like it’s isohedral (https://en.m.wikipedia.org/wiki/Isohedral_figure), so I doubt that there’s a justification for it being fair in principle.

henryseg,
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@wrog @richardelwes We often prototype our dice with 3D printing, but the final production dice are injection molded. Being convex (or almost convex) helps a lot with how possible a shape is to injection mold...
I don't exactly have the picture of the shape you are suggesting, but another of the issues is that if the space needed to write a number is small, the die needs to be correspondingly big.

gregeganSF, to random
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Take this polygon, and glue together edges with the same label, so the arrows point the same way. The heavy black lines are on the boundary of the resulting surface.

What is the topology of this space?

It turns out 2-dim manifolds can be fully classified by 3 things:

• Is the surface orientable, like a sphere or a torus, or not, like a Möbius strip?
• How many loops make up its boundary?
• Its rank: the max no. of non-intersecting curves joining 2 points on the boundary that don’t separate the space.

There’s a Möbius strip inside this space that you can find by drawing a quadrilateral that has the two edges labelled E as two opposite sides, so the space as a whole must be non-orientable.

The boundary forms a single closed loop, because both heavy black lines start and end at the same 2 points.

henryseg,
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@gregeganSF @OscarCunningham @caten @johncarlosbaez Yes, each P term reduces the Euler characteristic by one, starting from the sphere which has Euler characteristic two. Therefore P#P has Euler characteristic zero and it’s non orientable since it has a P term at all, so it is the Klein bottle by the classification theorem for compact surfaces (https://en.m.wikipedia.org/wiki/Surface_(topology)).

stecks, to random
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If anyone was thinking about applying for Matt Parker's MEGA Maths Engagement grant, the current round of applications will close on approximately 22/7 so get your applications in this week! Details: https://mathstodon.xyz/@standupmaths/110073123510203909

henryseg,
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@stecks The classic 22/7 approximation!

henryseg, to random
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Booth all set up at http://opensauce.live with @sabetta

henryseg,
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@sabetta Yesterday we were on our feet talking to people pretty much the whole day. Exhausting but great meeting so many people! Looking forward to the same thing today.

henryseg,
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@sabetta Not much time to look at other booths, but this was very cool, from https://www.planarmotor.com

video/mp4

monsoon0, to mathematics
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It seems to me that and is so important in , but this need constantly breaks against the language and terminology specific to and embedded in each subfield…

henryseg,
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@monsoon0 And there’s far worse than “blowup”. One of my favourite overloaded terms is “normal”.

henryseg,
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@monsoon0 At least “blowup” and “integrable” tell you something about what the term might mean. “Normal” is almost as generic as “good”, which I’ve only seen used somewhat sheepishly. Surely nobody calls something “good” and thinks anyone else will use their meaning of the term?

christianp, to 3DPrinting
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Stages in the design of a 3d-printed part. I did some very thin slices first, to check I'd got the shape right, then thought about how the whole thing should look.
Bonus points if you can guess what it's for!

henryseg,
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@christianp @xgranade I’ve also moved over to printables.com.

henryseg, to random
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Peaucellier–Lipkin linkage, made by my brother Will and me.

A 3D printed linkage draws a straight line with a pencil.

henryseg,
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@11011110 Yeah, turns out that there are easier ways to draw straight lines! Although I think historically these kinds of mechanisms were in the conversation on trying to improve the precision machining of flat surfaces.

henryseg,
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@11011110 I don't know this stuff well, but yeah I think that the original point was to make better steam engines. Then it turned out that the approximate solutions were good enough for the steam engines but by then I guess the mathematicians had got interested.

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