Henry Segerman (born 1979 in Manchester, UK) is a Professor of mathematics at Oklahoma State University in Stillwater, Oklahoma who does research in three-dimensional geometry and topology, especially three-manifolds, triangulations and hyperbolic geometry."Mathematical dice design and Variants of the 15-puzzle and the effects of holonomy" Kansas State University: Department of Mathematics
He was the first person to publish a book on mathematical 3D printing,Visualizing Mathematics with 3D Printing Johns Hopkins University Press (2016) and is also a recreational mathematician and mathematical artist with expertise in virtual reality.IVRPA Profile: Henry Segerman International Virtual Reality Professionals Association (IVRPA) His frequent collaborators include Vi Hart, Elisabetta Matsumoto and Saul Schleimer."Pumpkin geometry: stunning shadow sculptures that illuminate an ancient mathematical technique" by Alex Bellos, The Guardian, 2014-10-30Ornes, Stephen; Math Art: Truth, Beauty, and Equations Sterling Publishing (2019), "Projections" (Section 14)
Education and career Segerman received his Master of Mathematics (MS) at the University of Oxford (2001) and then his PhD at Stanford University (2007) for the dissertation "Incompressible Surfaces in Hyperbolic Punctured Torus Bundles are Strongly Detected" under Steven Paul Kerckhoff.Henry Segerman at the Mathematics Genealogy Project
He was a Lecturer at the University of Texas at Austin (2007–2010) and was a Research Fellow at University of Melbourne (2010–2013). He became an Assistant Professor at Oklahoma State University (2013–2018), then an Associate Professor (2018-2025), and then a full Professor from July 2025 to the present.Henry Segerman, The Bridges Organization: Mathematical Art Galleries, Oklahoma State University: Research Faculty Directory
Segerman's research lends itself to mathematics with a strong visual component. This led directly to his involvement with 3D printing. In 2016 he wrote the book Visualizing Mathematics With 3D Printing."Visualizing Mathematics With 3D Printing by Henry Segerman" reviewed by Laura Taalman, The American Mathematical Monthly, 22 Mar 2018, pp. 379-384 Laura Taalman in a review said, "Segerman's book is an inside tour of mathematics with breathtaking 3D-printed scenery."
3D printing
Mathematicians used to rely on wooden or plaster models to visualize complex geometrical shapes. Nowadays, if they can be described mathematically, we can "print" them with 3D printers."Can’t Imagine Shapes in 4 Dimensions? Just Print Them Out" By Luke Whelan, Wired, Nov 15, 2016 Segerman uses mathematical tools including quaternions,"Nothing Is More Fun than a Hypercube of Monkeys" By Evelyn Lamb, Scientific American, May 19, 2014 Euclidean and non-Euclidean geometries and stereographic projection to create instructions for 3D printers. Sometimes the goal is to produce a work of art."Fresh Off the 3D Printer: Henry Segerman’s Mathematical Sculptures" By Megan Gambino, Smithsonian Magazine, March 15, 2013 Sometimes it is to shed light on four-dimensional geometry or some other field such as mathematical group theory. Sometimes it is both. One of his sculptures depicts a set of monkeys joined together to form a 4-dimensional hypercube."Artist Uses 3-D Printing to Create 'Shadows' of 4-D Objects" NBC News: Weird Science, February 16, 2015,
Segerman's techniques help us visualize a four dimensional world."3D printing mathematics by Saul Schleimer and Henry Segerman, Plus Magazine, Millennium Mathematics Project Just as the frame of a cube can cast a shadow on a flat wall, Segerman makes analogous shadows of four dimensional objects via a 3D printer."Mathematicians at Play: 3-D Printing Enters the 4th Dimension" By Evelyn Lamb, Scientific American, October 31, 2012"Sculptures cast shadows from the fourth dimension" By Aviva Rutkin, New Scientist, 15 February 2015 Segerman has also explored kinetic artwork, designing mechanisms that move in unusual or seemingly paradoxical ways.
Recreational mathematics Segerman has appeared as a recreational mathematicianThe Dice You Never Knew You Needed by Siobhan Roberts, The New Yorker, 26 April 2016 at Gathering 4 Gardner conferencesHenry Segerman Gathering4Gardner and is a frequent contributor to Numberphile.A Layman’s Guide to Recreational Mathematics Videos Sam Enright
Combining his interests in mathematics and art he is one of 24 mathematicians and artists who make up the Mathemalchemy Team.Mathemalchemy’s Team
In another foray into recreational mathematics Segerman founded Dice Lab with mathematical artist Robert Fathauer. Using computer search and help from fellow recreational mathematician Robert Bosch, they created a "numerically balanced" 120-sided die in the shape of a disdyakis triacontahedron. It is the "biggest, most complex fair die possible".The Mathematics of Various Entertaining Subjects volume 2 European Mathematical Society They concede that the die is "expensive and there’s no real use for it", but it's still theoretically interesting.The Mind-Boggling Challenge of Designing 120-Sided Dice Wired Magazine, May 10, 2016
Selected papers Segerman does research in three-dimensional geometry and topology. Papers published in this area include: 2020 "Ray-marching Thurston geometries", arXiv:2010.15801 [math.GT, cs.GR, math.DS], [with Coulon, Matsumoto and Trettel], October 2020 2010 "Veering triangulations admit strict angle structures" [with Hodgson, Rubinstein and Tillmann] Geometry & Topology, November 2010, 15(4) 2015 "1-efficient triangulations and the index of a cusped hyperbolic 3-manifold", [with Garoufalidis, Hodgson and Rubinstein], Geometry & Topology 19 (2015), pp. 2619–2689
A second major interest, with some overlap, is mathematical visualization & art. Papers published in this area include: 2014 "The Quaternion Group as a Symmetry Group", [with Vi Hart], Proceedings of Bridges 2014: Mathematics, pp. 143–150 2012 "3D printing for mathematical visualisation", Math. Intell. 34(4) December 2012, pp. 56–62
Awards The paper "The Quaternion Group as a Symmetry Group" (with Vi Hart) was republished in The Best Writing on Mathematics 2015.The Best Writing on Mathematics 2015 Edited by Mircea Pitici, Princeton University Press (2022), The video "Non-Euclidean Virtual Reality Using Ray-Marching was a winner in the 2019 We Are Mathematics Video Competition.2019 Winners We Are Mathematics Video Competition!
References
External links Henry Segerman website