Henry Segerman

Henry Segerman (born 1979 in Manchester, UK) is an Associate 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.[1]

Henry Segerman at the Institute for Computational and Experimental Research in Mathematics in 2019
Photo by Sabetta Matsumoto

He was the first person to publish a book on mathematical 3D printing,[2] and is also a recreational mathematician and mathematical artist with expertise in virtual reality.[3] His frequent collaborators include Vi Hart, Elisabetta Matsumoto and Saul Schleimer.[4][5][6]

Education and careerEdit

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.[7]

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) and then an Associate Professor there in 2018 until the present.[5]

Segerman's research lends itself to mathematics with a strong visual component.[6] This led directly to his involvement with 3D printing. In 2016 he wrote the book Visualizing Mathematics With 3D Printing.[8] Laura Taalman in a review said, "Segerman's book is an inside tour of mathematics with breathtaking 3D-printed scenery."[8]

3D printingEdit

Triple gear
by Saul Schleimer and Henry Segerman
Figure eight knot complement
by François Guéritaud, Saul Schleimer, and Henry Segerman

Mathematicians used to rely on wooden or plaster models to visualizing complex geometrical shapes. Nowadays, if they can be described mathematically, we can "print" them with 3D printers.[9] Segerman uses mathematical tools including quaternions,[10] Euclidean and non-Euclidean geometries and stereographic projection to create instructions for 3D printers.[4] Sometimes the goal is to produce a work of art.[11] Sometimes it is to shed light on four-dimensional geometry[12] or some other field such as mathematical group theory. Sometimes it is both.[10] One of his sculptures depicts a set of monkeys joined together to form a 4-dimensional hypercube.[13]

Segerman's techniques help us visualize a four dimensional world.[14] 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.[15][12] Segerman has also explored kinetic artwork, designing mechanisms that move in unusual or seemingly paradoxical ways.[6]: 128 

Recreational mathematicsEdit

Segerman has appeared as a recreational mathematician[16] at Gathering 4 Gardner conferences[17] and is a frequent contributor to Numberphile.[18]

Combining his interests in mathematics and art he is one of 24 mathematicians and artists who make up the Mathemalchemy Team.[19]

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. It is the "biggest, most complex fair die possible".[20] They concede that the die is "expensive and there’s no real use for it", but it still theoretically interesting.[21]

Selected papersEdit

Segerman does research in three-dimensional geometry and topology. Papers published in this area include:

A second major interest, with some overlap, is mathematical visualization & art. Papers published in this area include:



  1. ^ "Mathematical dice design and Variants of the 15-puzzle and the effects of holonomy" Kansas State University: Department of Mathematics
  2. ^ Visualizing Mathematics with 3D Printing Johns Hopkins University Press (2016) ISBN 978-1421420356
  3. ^ IVRPA Profile: Henry Segerman International Virtual Reality Professionals Association (IVRPA)
  4. ^ a b "Pumpkin geometry: stunning shadow sculptures that illuminate an ancient mathematical technique" by Alex Bellos, The Guardian, 2014-10-30
  5. ^ a b Henry Segerman, The Bridges Organization: Mathematical Art Galleries
  6. ^ a b c Ornes, Stephen; Math Art: Truth, Beauty, and Equations Sterling Publishing (2019), "Projections" (Section 14) ISBN 978-1454930440
  7. ^ Henry Segerman at the Mathematics Genealogy Project
  8. ^ a b "Visualizing Mathematics With 3D Printing by Henry Segerman" reviewed by Laura Taalman, The American Mathematical Monthly, 22 Mar 2018, pp. 379-384
  9. ^ "Can’t Imagine Shapes in 4 Dimensions? Just Print Them Out" By Luke Whelan, Wired, Nov 15, 2016
  10. ^ a b "Nothing Is More Fun than a Hypercube of Monkeys" By Evelyn Lamb, Scientific American, May 19, 2014
  11. ^ "Fresh Off the 3D Printer: Henry Segerman’s Mathematical Sculptures" By Megan Gambino, Smithsonian Magazine, March 15, 2013
  12. ^ a b "Sculptures cast shadows from the fourth dimension" By Aviva Rutkin, New Scientist, 15 February 2015
  13. ^ "Artist Uses 3-D Printing to Create 'Shadows' of 4-D Objects" NBC News: Weird Science, February 16, 2015,
  14. ^ "3D printing mathematics by Saul Schleimer and Henry Segerman, Plus Magazine, Millennium Mathematics Project
  15. ^ "Mathematicians at Play: 3-D Printing Enters the 4th Dimension" By Evelyn Lamb, Scientific American, October 31, 2012
  16. ^ The Dice You Never Knew You Needed by Siobhan Roberts, The New Yorker, 26 April 2016
  17. ^ Henry Segerman Gathering4Gardner
  18. ^ A Layman’s Guide to Recreational Mathematics Videos Sam Enright
  19. ^ Mathemalchemy’s Team
  20. ^ The Mathematics of Various Entertaining Subjects volume 2 European Mathematical Society
  21. ^ The Mind-Boggling Challenge of Designing 120-Sided Dice Wired Magazine, May 10, 2016
  22. ^ The Best Writing on Mathematics 2015 Edited by Mircea Pitici, Princeton University Press (2022), ISBN 978-0691169651
  23. ^ 2019 Winners We Are Mathematics Video Competition!

External linksEdit