Crooked Houses: Visualizing the Polychora with Hyperbolic Patchwork

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Taneli Luotoniemi: Crooked Houses: Visualizing the Polychora with Hyperbolic Patchwork. In: Bridges 2017, Pages 17–24.



This paper presents kinetic models based on the 4-dimensional regular polytopes. The sequential ‘flattening’ is realized through the use of hyperbolic patchwork surfaces, which portray the bitruncated versions of the polychora. As pedagogical tools, these models offer a hands-on experience of 4D geometry.

Extended Abstract


 author      = {Taneli Luotoniemi},
 title       = {Crooked Houses: Visualizing the Polychora with Hyperbolic Patchwork},
 pages       = {17--24},
 booktitle   = {Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture},
 year        = {2017},
 editor      = {David Swart, Carlo H. S\'equin, and Krist\'of Fenyvesi},
 isbn        = {978-1-938664-22-9},
 issn        = {1099-6702},
 publisher   = {Tessellations Publishing},
 address     = {Phoenix, Arizona},
 note        = {Available online at \url{}}

Used References

[1] H. S. M. Coxeter. Regular Polytopes. Dover Publications. 1973.

[2] Robert A. Heinlein, “…And He Built a Crooked House”, In Astounding Science Fiction, February 1941.

[3] Saul Scleimer and Henry Segerman. Sculptures in S³. In Robert Bosch, Douglas McKenna, and Reza Sarhangi, editors, Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture, pages 103–110, Phoenix, Arizona, 2012, Tessellations Publishing. Available online at

[4] Carlo H. Séquin. 3D Visualization Model of the Regular Polytopes in Four and Higher Dimensions. In Reza Sarhangi, editors, Bridges: Mathematical Connections in Art, Music, and Science, pages 37–48, Southwestern College, Winfield, Kansas, 2002, Bridges Conference. Available online at

[5] Jeffrey R. Weeks. How To Sew a Hyperbolic Blanket. (as of April 21, 2017)

[6] Jeffrey R. Weeks. The Shape of Space. CRC Press. 2001.

[7] Wikipedia Contributors. Uniform 4-polytope. (as of April 21, 2017)


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