Henry SegermanA 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.
Cohomology fractal zoomHenry Segerman2019-12-17 | 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/.Gear cube and Brain gearHenry Segerman2022-12-28 | 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:50716 The earliest Brain gear I know of is this one, uploaded by mappum: thingiverse.com/thing:1532
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.
I used a Mini Maglite AAA LED Flashlight (in "candle mode") for this video. Cellphone flashlights also work well.Impossible trianglesHenry Segerman2022-10-09 | Exploring some ways to create the illusion of an impossible triangle.
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 OutroCannon-Thurston maps: naturally occurring space-filling curvesHenry Segerman2022-07-29 | 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.
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 CarvingsThe pi/4 polyhedronHenry Segerman2022-07-03 | Matthias Goerner's 3D print: http://shpws.me/SZbN 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
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).Countdown d24s #shortsHenry Segerman2022-06-19 | Full video at youtu.be/U0soSn7BojQWhy dont Rubiks cubes fall apart?Henry Segerman2022-05-09 | 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:
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 puzzleHenry Segerman2022-02-06 | 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.com Satellite beeps modified from a public domain recording of Sputnik, available at commons.wikimedia.org/wiki/File:Sputnik_beep.oggPuzzling degrees of freedomHenry Segerman2022-01-29 | 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.
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.)Where do these circles come from?Henry Segerman2022-01-01 | Shadertoy demo: shadertoy.com/view/7ds3zB 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.
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.Mathemalchemy buildHenry Segerman2021-08-03 | The main build of the Mathemalchemy project took place at Duke University in July 2021.
Thanks to Chaim Goodman-Strauss for the sculpture, to Saul Schleimer for naming the "rook", and to Sabetta Matsumoto for helpful conversations.MathemalchemyHenry Segerman2021-05-01 | For more details on the Mathemalchemy project, see mathemalchemy.orgWhy the 14-15 puzzle is impossible, and how to solve it anywayHenry Segerman2021-03-21 | A bit of the math of the 15-puzzle, and some variants that shake up that math.15 + 4 puzzleHenry Segerman2021-03-13 | 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:
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)Slant d6Henry Segerman2019-12-13 | Available at http://mathartfun.com/DiceLabDice.htmlBraiding gearsHenry Segerman2019-11-27 | 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.
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)Gripping gears with pass-through holesHenry Segerman2019-10-10 | A variant of gripping gears (youtu.be/ENFXnNtd3xU) adds holes so that a solid object can pass through the connection between the gears.
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)Checking the overs and unders of a knot on the London UndergroundHenry Segerman2019-09-23 | Matt Parker's video: youtu.be/b9OEuhdM6t8 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.Illustrating Geometry and Topology at ICERMHenry Segerman2019-09-17 | 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 gearsHenry Segerman2019-09-13 | 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 jitterbugsHenry Segerman2019-07-20 | 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.pdfSkew d8Henry Segerman2019-06-29 | Available at http://mathartfun.com/DiceLabDice.htmlTranslucent amber diceHenry Segerman2019-04-23 | These translucent amber dice are available from thedicelab.comHow many holes does a straw have?Henry Segerman2019-04-10 | A topologist's view on how to answer this question.
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 marchingHenry Segerman2019-03-01 | Non-euclidean virtual reality using ray marching
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.Extensor™ Construction Kit: InstructionsHenry Segerman2018-10-30 | This is the instructions video for the MathMechs™ extensor construction kit. For more information see http://mathmechs.com.