# What are some mathematical sculptures?

Either intentionally or unintentionally. Include location and sculptor, if known.

• Coincidentally or not, an article on mathematical sculpture just came out in the AMS Notices: ams.org/notices/201007/rtx100700840p.pdf – Charles Staats Jul 19 '10 at 13:24
• There is absolutely no reason to close this beautiful (literally) question. – Gil Kalai Nov 14 '13 at 21:13

The Mathematical Research Institute in Oberwolfach have a sculpture on their grounds depicting Boy's surface:

(source: mfo.de)

• It was a gift of the Mercedes-Benz... but is there an artist who made it? – Pietro Majer Jul 19 '10 at 13:47
• I don't read German, perhaps in the linked article here mfo.de/general/boy there's an answer? – Willie Wong Jul 19 '10 at 13:50
• The article seems to say that it was produced by a computer-aided-manufacturing system at Mercedes-Benz, rather than an artist. – user5117 Jul 20 '10 at 10:01
• so, remarkably, the most appeciated sculpture here was made by a computer? – Pietro Majer Jul 20 '10 at 16:34
• Well, many sculptures, even when "made by an artist" were in fact made after the artist programmed it into a computer, which then did the work. We don't say that other works were "made by a chisel" or "made by a paintbrush" do we? – Gerald Edgar Mar 13 '11 at 13:21

The centrepiece of McAllister building, which houses the math department at Penn State, is the Octacube, designed by Adrian Ocneanu.

There's a bit of a description of the mathematics behind the Octacube on the Penn State website, but unfortunately, that's the most material that I can find online. There are all kinds of animations set up to display on a computer terminal in McAllister building, but they don't seem to be available online anymore.

Very briefly, the mathematics of the sculpture is as follows: consider the four-dimensional regular convex polytope whose vertex set is the union of the vertex sets for the four-dimensional cube {(±1,±1,±1,±1)} and the four-dimensional octahedron {(±2,0,0,0), (0,±2,0,0), (0,0,±2,0), (0,0,0,±2)}. Consider the 1-skeleton of this polytope (vertices and edges), and project radially to S3 ⊂ ℝ4. Project the resulting "inflated polytope" stereographically to ℝ3, and "fatten" the edges so that a cross-section of an edge is no longer just a point, but a Y-shape (see the corners of the sculpture). What you get is the sculpture shown.

• Do you mean "project radially to $S^3$"? – Charles Staats Jul 19 '10 at 18:53
• Oops... I absolutely do. Fixed it in the post... thanks for the correction. – Vaughn Climenhaga Jul 19 '10 at 19:23

Helaman Ferguson. My department has one of these in the main office:

(source: helasculpt.com)

(source: memenet.or.jp)

• He's got some nice sculptures at MSRI, too. – Allen Knutson Oct 26 '10 at 0:54
• There is also a paper, Friedman and Sequin, Keizo Ushio’s Sculptures, Split Tori and Möbius Bands, available at maths.ed.ac.uk/~aar/papers/keizo.pdf – Gerry Myerson Nov 10 '13 at 23:09

Robert Longhurst made some surfaces as wood carvings.

MoMath, in partnership with Make, has a regular feature on DIY math sculptures. I personally like the space filling curve made of steel pipe ells by Chaim Goodman-Strauss and Eugene Sargent.

A model of the Riemann zeta function suggesting the zeroes and pole. One of several models at the University of Regensburg Math Dept.

3-D Permutahedron Sundial by Stefano Buonsignori (16th century) in the Medici collection presented by Museo Galileo. The 3-D permutahedron:

(The combinatorics underlying the geometry of the permutahedra are related to multiplicative inversion and, consequently, to Koszul duality and the formalism of Appell sequences and associated matrices, so I find this particular jewel at the juncture of art, engineering, geometry, combinatorics, and analysis particularly exciting. If only he had done one for the associahedron!)

Calabi-Yau space: the shape of the inner space of the universe. In Yau's hometown in south China.

See 3-D animation in "Where math meets physics" by Brockmeier, also intro in "Hidden dimensions" by Freiberger and the survey "The Calabi-Yau Landscape:from Geometry, to Physics, to Machine-Learning" by Yang-Hui He.

Eversion of a sphere: Models in resin made by Stewart Dickson of clay models made by Bernard Morin (a blind topologist!) of different stages of the eversion of a sphere. The resin models were presented at the Maubeuge symposium "Arts et Mathématiques." (Photo by John Sullivan.)

Excerpt (p. 146) from "Poincare's Prize" by G Szipiro:

(Smale) proved a theorem that showed it is possible to evert the sphere. But the procedure implicit in his theorem was so complicated that nobody could visualize it. Smale, and others who tried, wanted to see with their own eyes how the sphere would evert. For once, the ability to visualize may actually have been a handicap. It was left to the blind mathematician Bernard Morin to devise a procedure to turn a sphere inside out that could actually be implemented (if the membrane is able to pass through itself, that is.)

A sculpture by Gideon Weisz of an approximation of Alexander's horned sphere to five levels.

(Photo from “The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes” by David Darling.)

'Tripartite unit' [Unidade Tripartida], Max Bill

This sculpture is topologically equivalent to the connected sum of three projective planes with a point removed, corresponding to the only boundary component of the represented surface.

Isometric transformation between a catenoid and helicoid

Presented at the 2019 Mathematical Art Exhibition held at the AMS Joint Mathematical Meetings in Baltimore, MD. (30 x 25 x 25 cm, paper, mdf, steel, magnet, 2018.)

To the left is a paper-crafted catenoid, and to the right, a helicoid. These surfaces can be transformed isometrically into each other, i.e., without any strain, because the first fundamental form of the surfaces are identical. I made this model by weaving together paper strips whose shapes were obtained by minimizing the strain energy defined on a Riemannian manifold. The optimization problem was solved numerically using NURBS-based isogeometric analysis.---Yuto Horikawa

Dandelin spheres (description)

Clebsch diagonal surface (description)

The two models above are part of the M. Schelling Collection of the Geometric Models Collection at V. N. Karazin Karkiv National Univ. (See historical notes for the collection and in comments below.)

Fermi surface of copper with permutohedral Brillouin zone

As Kaganov and Lifshitz put it, the Fermi surface is‘the stage on which the ‘drama of the life of the electron’ is played out. -- Dugdale from Life on the edge: a beginner’s guide to the Fermi surface

See also "The Magic Wand Theorem of A. Eskin and M. Mirzakhani" by Zorich

• For more info on the instrument, see hsm.stackexchange.com/questions/5573/…. – Tom Copeland Jan 14 '17 at 0:49
• On the horned sphere youtube.com/watch?v=d1Vjsm9pQlc – Tom Copeland Feb 22 '19 at 0:47
• H. Weyl: Klein put a special enphasis on geometrical and physical intuition, which he managed to develop in the students by letting them construct solid (plaster, or metal) models of curves and surfaces in ordinary 3-space, or letting them draw very broad (1 x 2 meters) paper tables of cubic plane curves with an explicit plotting of their Q-rational points. A concrete witness to this tradition is the exhibition of plaster models of surfaces which are to be found still nowadays in the Halls of the Mathematical Institute in Gottingen. – Tom Copeland Feb 5 at 22:39
• (cont.) These models were then produced by the publishing company L. Brill in Darmstadt, later by the Schilling publishing company, and sold around the world: I have personally seen many of those in most of the older Deparments I have visited (cf. the 2 Volumes edited by G.Fischer on ”Mathematical Models” ... . // Writes G. Fischer : There were certainly other reasons than economic for the waning interest in models. ... . More and more general and abstract viewpoints came to the forefront of mathematics ... . Finally Nicolas Bourbaki totally banned pictures from his <their> books. – Tom Copeland Feb 5 at 22:41
• @nomen, lol, maybe that's the underlying, motivating sentiment of the Bourbakists. – Tom Copeland Feb 5 at 23:28

Adding to the list two of my favorite mathematical sculptors:

George Hart:

http://www.georgehart.com/sculpture/sculpture.html

Bathsheba:

http://www.bathsheba.com/

Finally, there are a lot of nice things at the new Geometry Playground in the Exploratorium, for anyone coming through the SF bay area:

http://www.exploratorium.edu/geometryplayground/

• Comments about and samples of George Hart's fascinating work can be found here: richbugger.wordpress.com/2009/12/04/… – Joseph Malkevitch Jul 20 '10 at 1:09
• +1 for George Hart; I had the pleasure of taking a few of his classes at Stony Brook. He's a great teacher, and a great person. Also, mathematical sculptures are his life. – BlueRaja Jul 20 '10 at 6:04

A month ago our team completed, over four days, a very large geometric sculpture out of 20 tons of snow. Eva Hild of Sweden designed it and came over to work with us. The complete story is at

http://stanwagon.com/snow/breck2011/index.html

The videos linked at the top of the page allow you to walk around the work.

Stan Wagon

• Wow! Very impressive. Shame that it will probably be left to melt (although the evolution of that might also be cool). – Todd Trimble Feb 8 '14 at 12:47

Borromean rings on campus of George Washington University in Washington, DC. Unfortunately, I didn't make a note of the name of the artist.

• The link is now broken. – KConrad Dec 16 '18 at 0:37
• @KCo, I'll try to find it on my computer. Meanwhile, I found a pictur of it on the web. – Gerry Myerson Dec 16 '18 at 8:33

I suspect that the Octacube is also the only mathematical sculpture (possibly the only mathematical topic?) to appear in the journal Playboy (March 2006).

Batsheba Grossman

(displayed this one here on MO before)

at Ohio State ...

The Garden of Constants is a sculptural garden of large numerals that highlight the activities performed in nearby College of Engineering buildings. The installation, by Barbara Grygutis, includes a black walkway featuring 50 individual formulas cast in bronze and embedded in handmade pavers. The Garden of Constants is on the lawn of Dreese Laboratories.

Here we see 6 and 8. The perfect number is golden colored.

• That settles the dispute whether 1 is prime! – Victor Protsak Jul 20 '10 at 3:36
• Victor, there are other numbers there besides 1 and primes, actually. They're just out of view. – KConrad Jul 20 '10 at 4:50
• Which is the "largest" number? – Douglas S. Stones Nov 18 '10 at 22:07
• This particular work consists only of the ten digits. – Gerald Edgar Mar 13 '11 at 13:23
• Taking "number field" literally. – Noam D. Elkies Nov 11 '13 at 21:39

I think tensegrity sculptures are cool. Kenneth Snelson does a lot of these.

(source: well.com)

Alessandro Giorgi has made a bunch of statues concerned with the mathematics of juggling, e.g. 1 and 2.

• It seems a bit misleading to have the balls attached to circles. Real juggling balls follow parabolic trajectories...! – Tobias Fritz Nov 15 '13 at 14:50

Cliff Stoll makes Klein bottles and sells them too. He's made the "worlds biggest klein bottle". You can see that here.

A ruled surface seen at the Peggy Guggenheim collection in Venice (I believe it is by Antoine Pevsner):

• That's really nice. Do you happen to know the medium it's done in? I think my mother (who is a sculptor) would be intrigued. – Todd Trimble Feb 8 '14 at 12:45
• @ToddTrimble: Hi Todd, it had a "coppery" look, but not sure if it was mixed with anything else. – auniket Feb 19 '14 at 13:50

Here are some sculptures by Henry Segerman: https://plus.google.com/u/0/photos/102006004474081559466/albums/5946201022131183089

John Robinson

Among several of his sculptures, personally for me the neatest one -

(it's at the Newton Institute)

I like the 4 dimensional mathematical sculptures by Bathsheba Grossman, such as the 24-cell:

http://www.bathsheba.com/math/24cell/24cell_new_th.jpg

Are the cryptographic sculptures by Jim Sanborn -- Cyrillic Projector, etc. -- close enough to a "mathematical sculpture"?

http://www.ncarts.org/images/afsb_art/workpix/14.jpg

The MSRIs eightfold way

• Sculptor: Helaman Ferguson – S. Carnahan Nov 19 '10 at 3:02
• link is broken... – Gottfried Helms Dec 21 '18 at 1:29
• Although the picture in the post no longer works, I suppose that some photos can be found elsewhere. – Martin Sleziak Jul 14 '19 at 12:46

The sculptures of Morton C Bradley are not so widely known. He was originally an art conservator in Boston, but he also explored geometric shapes and color to create many sculptures. They are "a reflection of his fascination with the science of color, his admiration for traditional patterns, his exploration of mathematical designs." (Quote from web site below.) He never sold a single piece of work. At his death he donated his entire estate to Indiana University. The IU Art Museum is cataloging his work and arranging exhibits. They've created a web site http://www.iub.edu/~iuam/online_modules/bradley/ that discusses and displays some of his work. In comparison with other work already mentioned here, perhaps the most significant thing is his deep exploration of color in his sculptures.

I am by coincidence in Paris at this moment, attending a meeting (the first) of ESMA, the European Society for Math and Art. George Hart is giving a talk in two hours on his sculpture. The website of ESMA is here: http://mathart.eu/ and the program of the meeting here: http://mathart.eu/en/conf10program.html, and a web tour of the associated exhibition is here: http://mathart.eu/ihp10/index.html There was a talk yesterday about Stan Wagon's snow sculptures by John Sullivan, one of the team.

Stan Wagon (with various collaborators) is known to make snow sculptures that are mathematically pleasant.

The Found Math galleries at the MAA website include many mathematical sculptures.

Check these out...they spring into shape by creating parallel folds on an initial paper shape:

http://erikdemaine.org/curved/

Erik Demaine is a mathematician (computational geometer) at MIT.

• One cannot mention Erik Demaine without also noting Robert Lang. langorigami.com – Willie Wong Jul 19 '10 at 13:00
• Don't forget Martin Demaine. – Joel Reyes Noche Apr 7 '11 at 2:16

"Tucker's group of genus 2" by Duane Martinez and DeWitt Godfrey

http://www.colgate.edu/news/blog/archives/archivedisplay?nwID=5031

http://commons.wikimedia.org/wiki/File:Tucker%27s_Genus_Two_Group.jpg

Does the National Aquatics Centre in Beijing count? It illustrates the Weaire-Phelan structure, a recent (and the first) counterexample to Kelvin's conjecture.

Here's one of a large number of nice mathematical sculptures at the Science Museum in London.

This sculpture is made of 30 interlocking pieces and requires many hands to assemble or disassemble.

This sprinkler illustrates a theorem that a 3d solid can be constructed to cast any desired set of silhouettes (also illustrated on the cover of Gödel, Escher, Bach).

The last three were found on the three-dimensional geometry page of this wonderful collection.

Finally, a shameless plug of one of my own questions.

The french sculptor Bernar (sic) Venet: see http://images.math.cnrs.fr/Bernar-Venet-de-l-art-et-des.html

• "Commutative Operation, 2001" made me laugh ;) – Piotr Achinger Feb 7 '14 at 18:22
• I liked how the devotee of art was staring at "S Matrix Element, 2001". – Todd Trimble Feb 8 '14 at 12:42

The gömböc—as, for example, in this image—is a homogeneous convex solid with one stable, one unstable, and no neutral point of equilibrium on a horizontal plane. Its existence was conjectured by Vladimir Arnol'd in 1995 proved in 2006 by Gábor Domokos and Péter Várkonyi. Its manufacture requires great precision, and true gömböcs are not hand-made by individual artists.

• (Just checking: did you intend to write that true gömböcök are not hand-made by artists? It kind-of makes sense, though the opposite statement would also make sense, so this could be an oversight. I shy away from changing this, of course, and will only correct the spelling.) – Peter Heinig Mar 5 '18 at 8:34
• @PeterHeinig: (1) In English, it is normal to anglicize the spelling, pronunciation, and grammatical declension of foreign words. My preferred spellings of gomboc and its plural form gombocs were both edited to gömböc; but in your comment the plural is distinguished as gömböcök. (2) The manufacturing tolerance is one part per thousand over a convex shape defined by complicated mathematical formulas. So the gomboc is well suited to computer-guided machining and would be very difficult, if possible at all, for a human sculptor to make. – John Bentin Mar 5 '18 at 10:12
• Thanks. I am sorry, the missing 's' after the second 'gömböc' was simply my mistake. I'll correct it. The umlaut seems very usual even in the English literature on the subject, though, so I'll keep the umlaut. Thanks for clarifying that gömböcök are usually not manufactured by individuals. – Peter Heinig Mar 5 '18 at 10:39
• Here's a link en.wikipedia.org/wiki/File:G%C3%B6mb%C3%B6c_statue.jpg to a large sculpture in Budapest that you could include in your post and a nice paper "Geometry and self-righting of turtles" by Domokos and Varkonyit (citeseerx.ist.psu.edu/viewdoc/…) – Tom Copeland Sep 24 at 19:04

Most of the artistic work of the Italian Attilio Pierelli was inspired by mathematics and notably by the idea of representing the fourth dimension. You can see some pictures of his "hyperspaces" at his site: www.pierelli.it/