Henry Segerman
Powered triple gear
updated
The earliest Gear cube I am aware of is this one, uploaded by Emmett Lalish: thingiverse.com/thing:50716
The earliest Brain gear I know of is this one, uploaded by mappum: thingiverse.com/thing:1532
Our versions are available to print yourself and assemble with bolts here:
Gear cube: printables.com/model/310705-gear-cube
Brain gear: printables.com/model/315255-brain-gear
Unknot disguises: http://shpws.me/Tk5e
The idea to make optical illusions with knots came from a project one of my students, Austin Elliott, did for the "3D printing and math" class I teach at Oklahoma State. In Austin's design the knot could cast shadows in two directions, one of which looked like the shadow of a trefoil knot, and the other looked like the shadow of a figure eight knot.
The shadow of a knot loses the crossing information - in my versions you can see the crossing information (or at least, the crossing information of the disguise), but it seems harder to make the knot look nice and smooth from a side view as well as the "disguise" view.
Secret evil extra puzzle: Starting at 1:36, why is there a faint extra image of the knot in the mirror?
One tile: http://shpws.me/TiT7
Full frame: http://shpws.me/TiT6 (will also require three 25mm long M2 bolts and three M2 nuts)
20 tiles: http://shpws.me/TiT5
Files for the half frame: printables.com/model/323666-helix-cube-puzzle-prototype
Soon after, someone attached a click-bait caption to the image, and it has been bouncing around the internet ever since. Here, I explain what the image is really about.
You can buy these stereographic projection spheres from shapeways.com/shops/henryseg?section=Stereographic+Projection, or download and print yourself from printables.com/social/246633-henryseg/collections/309464
I used a Mini Maglite AAA LED Flashlight (in "candle mode") for this video. Cellphone flashlights also work well.
Impossible triangle (helix):
http://shpws.me/TggA
printables.com/model/292131-impossible-triangle-helix
tato_713's "Penrose triangle like figure"
printables.com/model/157441-penrose-triangle-like-figure
Impossible triangle (tilted planes):
http://shpws.me/TggB
printables.com/model/292132-impossible-triangle-tilted-planes
Buy from Shapeways: http://shpws.me/Te9N
Download to print yourself: printables.com/model/270848-squaring-the-circle-illusion
Read our paper: archive.bridgesmathart.org/2022/bridges2022-313.pdf
Instructions and files for printing and making one of these grids at printables.com/model/177955-kinetic-cyclic-scissors
This research was partially supported by the Koslow Undergraduate Mathematics Research Experience Scholarship.
00:00 Introduction
00:19 Self-similar quadrilateral tilings
00:45 Scissor grids
00:55 Can the grid always move?
01:44 What's going on in general
04:06 Cyclic quadrilaterals
04:36 Example: 6/12/8/9
05:08 Example: 1-1 path
06:16 Example: 1-2 path
07:02 Outro
Thanks to my brother Will Segerman for making the carvings, and to Daniel Piker for making the figure-eight knot animations. I made the animation of the (super crinkly) surface using our app (with Dave Bachman) for cohomology fractals. You can play with the app (on Chrome or Firefox) at henryseg.github.io/cohomology_fractals/.
Also see:
Cannon and Thurston, Group invariant Peano curves, Geom. Topol., 2007.
Mumford, Series, and Wright, Indra's pearls, Cambridge University Press, 2002.
Some of these curves are available in t-shirt form at neatoshop.com/artist/Henry-Segerman.
00:00 Introduction
00:28 The Hilbert curve
01:00 Approximations to Cannon-Thurston map
01:36 What space do they fill?
02:01 Symmetry of the Hilbert curve
02:34 Symmetry of the Cannon-Thurston map
03:10 The Hilbert curve is artificial
03:38 The complement of the figure-eight knot
04:39 The universal cover
05:20 Unwrapping the surface in the knot complement
05:51 The crinkling
06:50 Thurston's pictures
07:24 Comparing algorithms
08:23 s227
09:18 Carvings
Countdown d24: youtu.be/U0soSn7BojQ
Matthias' version of the construction of the polyhedron: http://www.unhyperbolic.org/sydler.html
Demonstration of the Wallace–Bolyai–Gerwien theorem by Dima Smirnov and Zivvy Epstein: dmsm.github.io/scissors-congruence
Brooks and Matelski were the first to make an image of what we now call the Mandelbrot set. This image is in the public domain: en.wikipedia.org/wiki/Mandelbrot_set#/media/File:Mandel.png
According to Wikipedia, Pingala studied the relations between the numbers in what we now call Pascal's triangle, but the first appearance of these numbers arranged in a triangle was due to the Persian mathematician Al-Karaji (953–1029).
It would be far cheaper to just buy a Rubik's cube and disassemble it, but in case you want to buy one of these 3D printed mechanisms, you can get the parts here:
Faces and core: shpws.me/T4WN
Edges: shpws.me/T4WQ
Corners: shpws.me/T4WR
You can also download the files from printables.com/model/201027-twisty-cube-mechanism and print the parts on your own printer.
Countdown d24: mathartfun.com/thedicelab.com/CountdownD24.html
Thanks to Alexandre Muñiz for the idea to make a countdown die based on a sphericon.
My article on Rolling Acrobatic Apparatus:
ams.org/journals/notices/202107/rnoti-p1106.pdf
You can 3D print your own sphericon and three-cone sphericon:
printables.com/model/177077-sphericon
printables.com/model/177213-three-cone-sphericon
Extensor kits are available from mathmechs.com.
The accessories to make an extensor grabber are available to download and 3D print for yourself at printables.com/model/168603-extensor-grabber. Alternatively, you can buy the extra parts from Shapeways at http://shpws.me/T3q0.
(Yes, I know, 3D printing is very expensive in comparison to mass-produced injection molded products. If you are a company interested in mass-producing any of my puzzles or mechanisms, please get in touch!)
Stickers available on request from http://www.chewiescustomstickers.com
Satellite beeps modified from a public domain recording of Sputnik, available at commons.wikimedia.org/wiki/File:Sputnik_beep.ogg
Thanks to Bram Cohen for asking a question that led to this puzzle.
Go check out Mr.Puzzle's YouTube channel (youtube.com/c/MrPuzzle). I am flattering his spoiler break screen extremely sincerely.
Shapeways: http://shpws.me/SY2F
The original Shadertoy demo: shadertoy.com/view/7ds3zB
The closest parallel planes:
Suppose that v is an integer vector with no common factors between its coordinates. We want to find the integer vector u so that the component of u in the direction of v is as small as possible but still strictly positive. This component can be written using the dot product as u·v/|v|. Since v has no common factors between its coordinates, we can find an integer vector u so that u·v = 1. Thus the distance between the plane perpendicular to v and based at the origin and the parallel plane based at the end of u is 1/|v|. There is no closer plane, since u.v must be an integer.
Correction: the ripple frequency is proportional to the length of v, not inversely proportional. (Longer vectors have higher frequency ripples.)
Thanks to Ravi Vakil (and his son Benjamin) for noticing the circles and asking the question, and to CodeParade (http://codeparade.net) for the photos of the Legoland globe.
Followup video answering the question about the rates of the ripples: youtu.be/yrDbD90HXyo
Thanks to Sabetta Matsumoto and Chaim Goodman-Strauss for building the sculpture, to Saul Schleimer for naming the "rook", and to all of them for helpful conversations.
More information about the project: mathemalchemy.org
Hyperbolic 29 Puzzle Tiles: http://shpws.me/SJra
Hyperbolic 29 Puzzle Frame: http://shpws.me/SJr9
Hyperbolic 12 Puzzle Tiles: http://shpws.me/SJrz
Hyperbolic 12 Puzzle Frame: http://shpws.me/SJry
Numbers to print (for either set of tiles): thingiverse.com/thing:4897366
HyperRogue simulations:
http://roguetemple.com/z/29
http://roguetemple.com/z/12
http://roguetemple.com/z/124
http://roguetemple.com/z/60
Thanks also to the makers of HyperRogue for the end screen animation!
Thanks to Chaim Goodman-Strauss for the sculpture, to Saul Schleimer for naming the "rook", and to Sabetta Matsumoto for helpful conversations.
If you want to buy one of these, you will need the following two prints from Shapeways:
Frame: http://shpws.me/SzQG
Tiles: http://shpws.me/SzQH
You will also need ten 8mm long M2 bolts and nuts to form the hinges of the puzzle. I used this set: amazon.com/DYWISHKEY-Pieces-Socket-Screws-Wrench/dp/B07W5J19Y5
A page describing Bishop cubes: sites.google.com/site/geduldspiele/PuzzleReviewBISHOPCUBES
The puzzle is available from Shapeways. You will need both the tiles and the frame parts, together with three M2 nuts and bolts that are at least 8mm long.
Frame: http://shpws.me/RQ27
Tiles: http://shpws.me/RQ28
Cohomology fractal zoom: youtu.be/-g1wNbC9AxI
Non-euclidean virtual reality using ray-marching: youtu.be/ivHG4AOkhYA
This is joint work with Dave Bachman and Saul Schleimer. This video was filmed during ICERM's Fall 2019 "Illustrating Mathematics" program, which was made possible through the support of
- The National Science Foundation (Grant Number DMS-1439786)
- The Alfred P. Sloan Foundation (Grant Number G-2019-11406)
- A Simons Foundation Targeted Grant to Institutes (Award ID 546235)
See our explanation video at youtu.be/fhBPhie1Tm0 for how we make these fractals.
Music: "Tossing and Turning" by Vi Hart, licensed under a CC BY-NC 3.0 license: creativecommons.org/licenses/by-nc/3.0/.
Shapeways 3D printed version: http://shpws.me/SXbE
This video was filmed during ICERM's Fall 2019 "Illustrating Mathematics" program, which was made possible through the support of
- The National Science Foundation (Grant Number DMS-1439786)
- The Alfred P. Sloan Foundation (Grant Number G-2019-11406)
- A Simons Foundation Targeted Grant to Institutes (Award ID 546235)
3D Print files at Thingiverse: thingiverse.com/thing:3908970
This video was filmed during ICERM's Fall 2019 "Illustrating Mathematics" program, which was made possible through the support of
- The National Science Foundation (Grant Number DMS-1439786)
- The Alfred P. Sloan Foundation (Grant Number G-2019-11406)
- A Simons Foundation Targeted Grant to Institutes (Award ID 546235)
Geoff Marshall's behind-the-scenes video: youtu.be/IIMbF_Amuc4
Beautiful diagrams of tube platform heights: dansilva.co.uk/down-underground
Sabetta Matsumoto and I went to check that the knot we made on the tube (with Matt Parker, Geoff Marshall and Vicki Pipe) was actually the left-handed trefoil, and not some other knot.
ICERM's Fall 2019 "Illustrating Mathematics" program was made possible through the support of
- The National Science Foundation (Grant Number DMS-1439786)
- The Alfred P. Sloan Foundation (Grant Number G-2019-11406)
- A Simons Foundation Targeted Grant to Institutes (Award ID 546235)
This video was filmed during ICERM's Fall 2019 "Illustrating Mathematics" program, which was made possible through the support of
- The National Science Foundation (Grant Number DMS-1439786)
- The Alfred P. Sloan Foundation (Grant Number G-2019-11406)
- A Simons Foundation Targeted Grant to Institutes (Award ID 546235)
http://archive.bridgesmathart.org/2019/bridges2019-399.pdf
en.wikipedia.org/wiki/Homotopy_group
en.wikipedia.org/wiki/Homology_(mathematics)
en.wikipedia.org/wiki/Persistent_homology
This variant jitterbug mechanism is based on the cuboctahedron, expanding to become a rhombicuboctahedron. For this mechanism the rotation rates of the triangular and square parts are not linearly related to each other. This means that our gears are not circular. Joint work with Sabetta Matsumoto.
Try out the simulation at http://michaelwoodard.net/hypVR-Ray
The code is available at github.com/mtwoodard/hypVR-Ray
Joint work with Roice Nelson and Michael Woodard.
This video demonstrates a virtual reality simulation of a non-euclidean, negatively curved space.
Suppose that I walk in a straight line, then turn left by 90 degrees, and then repeat these steps until I get back to where I started. In a flat space with no curvature, I have just walked around a square. In a positively curved space, like the surface of a sphere, I could instead have walked around a right-angled triangle: I could start on the equator and walk a quarter the way around the sphere, then turn left and go up to the north pole, then turn left again and walk down to my starting point. In this video I am (virtually) in a negatively curved space; here I walk around a right-angled pentagon.
The study of negatively curved spaces like this (more specifically, hyperbolic space) is an intense area of research in three-dimensional geometry and topology. This has grown from the seminal work of William Thurston, who showed that "most" three-dimensional spaces are hyperbolic - the flat, euclidean spaces are the unusual ones!
We created this simulation to make non-euclidean spaces more accessible to the general public: The simulation is available at michaelwoodard.net/hypVR-Ray and works in any web-browser - there is even a version for iOS and Android devices. In the future, we hope to implement simulations of other non-euclidean spaces, including some of the other geometries described by Thurston in his foundational "geometrization conjecture", which was proved by Grigori Perelman in 2003. We also plan to integrate our visualization work into software used by other researchers, such as the program "SnapPy", which is used to study hyperbolic manifolds.
Our simulation is programmed using ray-marching, a graphics technique similar to ray-tracing. For each pixel of the screen, the program decides how to colour the pixel by tracing a ray of light from the pixel out into the world of the simulation and seeing which object it hits. As described in the video, an implementation of ray-marching requires very little knowledge about the space we are simulating, only:
1) A way to write down points in the space (i.e. a model for the space),
2) signed distance functions for the objects in the world, and
3) a way to move a given distance along a ray.
This should help us generalise the technique to other, even weirder spaces.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
Parts: 0:27
Y-extensor: 1:20
Y-extensor (3 stages): 4:03
A trick for removing plugs: 5:00
X-extensor: 5:37
Extensor Slinger: 6:17
Caltrop: 8:08
Elbow: 12:48
Square: 13:27
Axes: 13:56
Cube: 14:24
Dodecahedron: 14:50
Diamond: 16:19
Tetris piece: 18:04