Further adventures in stereographic projection Henry Segerman 2026-09-18 | Further adventures in stereographic projection
Gear cube and Brain gear Henry Segerman 2022-12-28 | Exploring some mechanisms based on bevel gears, with Sabetta Matsumoto. These are our interpretations of some reasonably well known designs. The earliest Gear cube I am aware of is this one, uploaded by Emmett Lalish: thingiverse.com/thing:50716The earliest Brain gear I know of is this one, uploaded by mappum: thingiverse.com/thing:1532Our versions are available to print yourself and assemble with bolts here:Gear cube: printables.com/model/310705-gear-cubeBrain gear: printables.com/model/315255-brain-gear
Knots in disguise Henry Segerman 2022-12-08 | Trefoil disguises: http://shpws.me/Tk5wUnknot disguises: http://shpws.me/Tk5eThe 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?
Helix cube puzzle Henry Segerman 2022-11-20 | Prototype half frame: http://shpws.me/TiT4One tile: http://shpws.me/TiT7Full frame: http://shpws.me/TiT6 (will also require three 25mm long M2 bolts and three M2 nuts)20 tiles: http://shpws.me/TiT5Files for the half frame: printables.com/model/323666-helix-cube-puzzle-prototype
LIGHT shadow (stereographic projection) Henry Segerman 2022-10-30 | Stereographic projection spheres available at shapeways.com/shops/henryseg?section=Stereographic+Projection
This is not about spacetime Henry Segerman 2022-10-23 | I originally took this photo illustrating stereographic projection in 2014. 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/309464I used a Mini Maglite AAA LED Flashlight (in "candle mode") for this video. Cellphone flashlights also work well.
Impossible triangles Henry Segerman 2022-10-09 | Exploring some ways to create the illusion of an impossible triangle.Impossible triangle (helix): http://shpws.me/TggAprintables.com/model/292131-impossible-triangle-helixtato_713's "Penrose triangle like figure"printables.com/model/157441-penrose-triangle-like-figureImpossible triangle (tilted planes): http://shpws.me/TggBprintables.com/model/292132-impossible-triangle-tilted-planes
Genus two holonomy Henry Segerman 2022-09-12 | Available at shapeways.com/shops/henryseg?section=Holonomy+mazes
Squaring the circle illusion #shorts Henry Segerman 2022-09-04 | How to make a shape that looks like a square from one direction and a circle from another.Buy from Shapeways: http://shpws.me/Te9NDownload to print yourself: printables.com/model/270848-squaring-the-circle-illusion
Kinetic cyclic scissors Henry Segerman 2022-08-17 | Joint work with Kyle VanDeventer.Read our paper: archive.bridgesmathart.org/2022/bridges2022-313.pdfInstructions and files for printing and making one of these grids at printables.com/model/177955-kinetic-cyclic-scissorsThis research was partially supported by the Koslow Undergraduate Mathematics Research Experience Scholarship.00:00 Introduction00:19 Self-similar quadrilateral tilings00:45 Scissor grids00:55 Can the grid always move?01:44 What's going on in general04:06 Cyclic quadrilaterals04:36 Example: 6/12/8/905:08 Example: 1-1 path06:16 Example: 1-2 path07:02 Outro
Cannon-Thurston maps: naturally occurring space-filling curves Henry Segerman 2022-07-29 | Saul Schleimer and I attempt to explain what a Cannon-Thurston map is.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 Introduction00:28 The Hilbert curve01:00 Approximations to Cannon-Thurston map01:36 What space do they fill?02:01 Symmetry of the Hilbert curve02:34 Symmetry of the Cannon-Thurston map03:10 The Hilbert curve is artificial03:38 The complement of the figure-eight knot04:39 The universal cover05:20 Unwrapping the surface in the knot complement05:51 The crinkling06:50 Thurston's pictures07:24 Comparing algorithms08:23 s22709:18 Carvings
The pi/4 polyhedron Henry Segerman 2022-07-03 | Matthias Goerner's 3D print: http://shpws.me/SZbNCountdown d24: youtu.be/U0soSn7BojQMatthias' version of the construction of the polyhedron: http://www.unhyperbolic.org/sydler.htmlDemonstration of the Wallace–Bolyai–Gerwien theorem by Dima Smirnov and Zivvy Epstein: dmsm.github.io/scissors-congruenceBrooks 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.pngAccording 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).
Why dont Rubiks cubes fall apart? Henry Segerman 2022-05-09 | Explaining (one version of) the mechanism inside a Rubik's cube.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/T4WNEdges: shpws.me/T4WQCorners: shpws.me/T4WRYou can also download the files from printables.com/model/201027-twisty-cube-mechanism and print the parts on your own printer.
From Sphericons to Countdown dice Henry Segerman 2022-04-24 | How we made a better countdown/spindown die, inspired by sphericons.Countdown d24: mathartfun.com/thedicelab.com/CountdownD24.htmlThanks 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.pdfYou can 3D print your own sphericon and three-cone sphericon:printables.com/model/177077-sphericonprintables.com/model/177213-three-cone-sphericon
Grabber mechanism Henry Segerman 2022-04-16 | Featuring my brother Will Segerman, and filmed by Sabetta Matsumoto.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.
Continental drift puzzle Henry Segerman 2022-02-06 | Available at shapeways.com/shops/henryseg?section=Continental+Drift+Puzzle(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.comSatellite beeps modified from a public domain recording of Sputnik, available at commons.wikimedia.org/wiki/File:Sputnik_beep.ogg
Puzzling degrees of freedom Henry Segerman 2022-01-29 | Clarification: for the “two rotational degrees of freedom” question, the rotations should all fix a common center point. Otherwise the solution for two degrees of freedom is already a solution.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
More circles on a sphere of cubes Henry Segerman 2022-01-15 | This is a followup video to youtu.be/A2IAyXc0LuE.The original Shadertoy demo: shadertoy.com/view/7ds3zBThe 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.)
Where do these circles come from? Henry Segerman 2022-01-01 | Shadertoy demo: shadertoy.com/view/7ds3zBThanks 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
Holonomy mazes without a maze Henry Segerman 2021-09-06 | Available at shapeways.com/shops/henryseg?section=Holonomy+mazes
A better d6 than the cube? Henry Segerman 2021-08-29 | Available to buy from mathartfun.com/DiceLabDice.html
Dodecahedral holonomy maze Henry Segerman 2021-08-21 | Available from Shapeways: shapeways.com/shops/henryseg?section=Holonomy+mazesThanks 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.
Mathemalchemy build Henry Segerman 2021-08-03 | The main build of the Mathemalchemy project took place at Duke University in July 2021. More information about the project: mathemalchemy.org
Glow-in-the-dark d120 Henry Segerman 2021-07-18 | Available to buy from mathartfun.com/DiceLabDice.html
Hyperbolic 29 puzzle Henry Segerman 2021-07-01 | Buy from Shapeways:Hyperbolic 29 Puzzle Tiles: http://shpws.me/SJraHyperbolic 29 Puzzle Frame: http://shpws.me/SJr9Hyperbolic 12 Puzzle Tiles: http://shpws.me/SJrzHyperbolic 12 Puzzle Frame: http://shpws.me/SJryNumbers to print (for either set of tiles): thingiverse.com/thing:4897366HyperRogue simulations:http://roguetemple.com/z/29http://roguetemple.com/z/12http://roguetemple.com/z/124http://roguetemple.com/z/60Thanks also to the makers of HyperRogue for the end screen animation!
(Octahedral) holonomy maze Henry Segerman 2021-05-04 | Available from Shapeways: shapeways.com/shops/henryseg?section=Holonomy+mazesThanks to Chaim Goodman-Strauss for the sculpture, to Saul Schleimer for naming the "rook", and to Sabetta Matsumoto for helpful conversations.
Mathemalchemy Henry Segerman 2021-05-01 | For more details on the Mathemalchemy project, see mathemalchemy.org
Why the 14-15 puzzle is impossible, and how to solve it anyway Henry Segerman 2021-03-21 | A bit of the math of the 15-puzzle, and some variants that shake up that math.
15 + 4 puzzle Henry Segerman 2021-03-13 | The 15 + 4 puzzle is a hinged version of the classic 15 Puzzle, with an extra 4 tiles.If you want to buy one of these, you will need the following two prints from Shapeways:Frame: http://shpws.me/SzQGTiles: http://shpws.me/SzQHYou 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
Skew dice set Henry Segerman 2020-11-25 | The skew dice set is available at mathartfun.com/DiceLabDice.html.
Rook cubes (unsuccessful prototype) Henry Segerman 2020-04-25 | Oskar van Deventer's "Segerdas": youtube.com/watch?v=G_RlsIe3-qkA page describing Bishop cubes: sites.google.com/site/geduldspiele/PuzzleReviewBISHOPCUBES
Coiled 15 puzzle Henry Segerman 2020-03-30 | This is a variant of the classic sliding tile puzzle, where the puzzle is coiled around a cylinder. The goal is the same: to put the tiles back in the correct order. Unlike the classic 15-puzzle, this version is solvable because of the extra connection between the first and sixteenth squares of the frame.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/RQ27Tiles: http://shpws.me/RQ28
Cohomology fractals Henry Segerman 2019-12-20 | We explain how to make fractal images from cohomology classes in hyperbolic three-manifolds. You can try out the web app for yourself at: henryseg.github.io/cohomology_fractalsCohomology fractal zoom: youtu.be/-g1wNbC9AxINon-euclidean virtual reality using ray-marching: youtu.be/ivHG4AOkhYAThis 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)
Cohomology fractal zoom Henry Segerman 2019-12-17 | A zoom through a selection of cohomology fractals generated by henryseg.github.io/cohomology_fractals. In the web app there: arrow and wasd keys to move, choose the manifold from the dropdown menu. The names of the manifolds are as given in the SnapPea census.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/.
Braiding gears Henry Segerman 2019-11-27 | These three gears hold themselves together without an axle or a frame, like our "Gripping gears" (youtu.be/RBZG8M8_a8Y). But these can change how they are connected together - in fact they can "braid" around each other however you like.Shapeways 3D printed version: http://shpws.me/SXbEThis 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)
Gripping gears with pass-through holes Henry Segerman 2019-10-10 | A variant of gripping gears (youtu.be/ENFXnNtd3xU) adds holes so that a solid object can pass through the connection between the gears.3D Print files at Thingiverse: thingiverse.com/thing:3908970This 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)
Checking the overs and unders of a knot on the London Underground Henry Segerman 2019-09-23 | Matt Parker's video: youtu.be/b9OEuhdM6t8Geoff Marshall's behind-the-scenes video: youtu.be/IIMbF_Amuc4Beautiful diagrams of tube platform heights: dansilva.co.uk/down-undergroundSabetta 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.
Illustrating Geometry and Topology at ICERM Henry Segerman 2019-09-17 | This is a short video showcasing the activities on the first day (2019-09-16) of the Illustrating Geometry and Topology workshop at ICERM, the Institute for Computational and Experimental Research in Mathematics, at Brown University, RI.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)
Gripping gears Henry Segerman 2019-09-13 | Joint work with Will Segerman and Sabetta Matsumoto.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)
Bridges 2019 talk - Geared jitterbugs Henry Segerman 2019-07-20 | This is a talk I gave with Sabetta Matsumoto at the Bridges conference on mathematics and the arts (http://bridgesmathart.org), on 18th July 2019, about our paper: http://archive.bridgesmathart.org/2019/bridges2019-399.pdf
Translucent amber dice Henry Segerman 2019-04-23 | These translucent amber dice are available from thedicelab.com
How many holes does a straw have? Henry Segerman 2019-04-10 | A topologist's view on how to answer this question.en.wikipedia.org/wiki/Homotopy_groupen.wikipedia.org/wiki/Homology_(mathematics)en.wikipedia.org/wiki/Persistent_homology
Geared Cuboctahedral Jitterbug Henry Segerman 2019-03-24 | 3d print available at http://shpws.me/R8NY. 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.
Non-euclidean virtual reality using ray marching Henry Segerman 2019-03-01 | Non-euclidean virtual reality using ray marchingTry out the simulation at http://michaelwoodard.net/hypVR-RayThe code is available at github.com/mtwoodard/hypVR-RayJoint 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, and3) 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.
Extensor™ Construction Kit: Instructions Henry Segerman 2018-10-30 | This is the instructions video for the MathMechs™ extensor construction kit. For more information see http://mathmechs.com.Parts: 0:27Y-extensor: 1:20Y-extensor (3 stages): 4:03A trick for removing plugs: 5:00X-extensor: 5:37Extensor Slinger: 6:17Caltrop: 8:08Elbow: 12:48Square: 13:27Axes: 13:56Cube: 14:24Dodecahedron: 14:50Diamond: 16:19Tetris piece: 18:04
