Noble Polyhedra


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In 2026 Connor Hill stunned the world of mathematics with his paper "the complete set of noble polyhedra" [1] in which he provides "a complete enumeration of the finite polyhedra that are noble, that is, polyhedra that are vertex transitive and facet transitive".  In addition to those with dihedral symmetry, Hill produced a list  of 146 noble polyhedra.  This paper won Hill the top $250,000 prize at the Regeneron Science Talent Search.

The study of noble polyhedra, above and beyond the Platonic solids and the Kepler-Poinsot polyhedra, began with Hess in 1875 who discovered 16 new cases [2,3,4].  Bruckner then added a further 10 cases between 1900 and 1907 [5,6,7,8]. 

There matters rested until the advent of modern computer tools, Robert Webb found a new case in 2008 [9], Ulrich Milkloweit a further 33 in 2020 [10] and Stella user 'senkoquartz' a further 11 [11].  All using Webb's Stella software [12]. 

Hill's discovery not only added a significant number of cases, but also provided a proof that the list was complete and gave a methodology for determing the vertices of the polyhedra as algebraic solutions to polynomial equations.  I will not atttempt to repeat Hill's arguments here as they are fully set out in his paper.

I will though briefly describe my own journey of discovery:

Knowing of Hill's discoveries via the polytope wiki at polytope.miraheze.org [13] but prior to the publication of his paper, I was asked 'How do you go about finding a noble polyhedron which is not a faceting of a uniform polyhedron?' [Fn 1].  I considered the following:

1.  Place a vertex in a Schwartz triangle - for the icosahedral cases (π/5, π/2, π/3).
2.  For a noble polyhedron with a gonality of n, choose n vertices of the polyhedron (initially at random) but eventually a systematic search can be performed.
3.  Move the inital vertex point around in the Schwartz triangle (also adjusting the positions of all other vertices) until the chosen vertices are coplanar.
4.  Form polygons from the chosen vertices (there will be a number of closed cycles that will need to be tested)
5.  Test the resulting mesh to see if the faces form a valid polyhedron.

If all 5 steps can be successfuly completed, the resulting form will be a noble polyhedron.

To implement my ideas, I turned to Google AI Studio, and specifically Gemini.  One change to my logic was necessary though for practical purposes

1.  Place a vertex in a Schwartz triangle - for the icosahedral cases (π/5, π/2, π/3).
2.  For a noble polyhedron with a gonality of n, choose n vertices of the polyhedron (initially at random) but eventually a systematic search can be performed.
3.  Form polygons from the chosen vertices (there will be a number of closed cycles that will need to be tested)
4.  Test the resulting mesh topologically to see if the faces form a valid polyhedron.
5.  Move the initial vertex point around in the Schwartz triangle (also adjusting the positions of all other vertices) until the chosen vertices are coplanar.

It was far more efficeint to perform the 'topologic' test (the possibility of which I had not foreseen) before the more CPU intense coplanarity solver.

With the assistance of the AI I was able to produce python code that could produce all 146 of Hill's noble polyhedra.

I also generated a set of minimal symmetric injective colourings for the complete set of noble polyhedra—a result that, to the best of my knowledge, has not been previously documented.

My python code is available here.

A zip file containing OFF files for all noble polyhedra described on these pages is available here. 

A larger zip file containing OFF, WRL, X3D and X3DV files is available here.

I now present all of Hill's 146 noble polyhedra.  For practical purposes I have divided them into categories mainly based on gonality and symmetry.



 References

1  Hill, Connor "The complete set of noble polyhedra" (2026). arXiv. https://arxiv.org/pdf/2607.28711
 2. Edmund Hess. “zwei Erweiterungen des Begriffs der regelm¨assigen K¨orper. (German) [Two extensions to the concept of regular bodies]”. In: der Gesellshaft zur Bedf¨ordung der gesammten Naturwissenschaften (1875), pp. 3–26.
3. Edmund Hess. Ueber die zugleich gleicheckigen und gleichfl¨achigen Polyeder. (German) [About the simultaneously vertex-transitive and face-transitive polyhedra]. zwei Erweiterungen des Begriffs der regelm¨assigen K¨orper. Kay, 1876.
4. Edmund Hess. “Ueber einige merkw¨urdige nicht convexe Polyeder. (German) [About some strange nonconvex polyhedra]
5. Max Brückner. Vielecke und Vielflache: Theorie und Geschichte. (German) [Polygons and Polyhedra: Theory and History]. Teubner, 1900.
6. Max Brückner. “Uber die diskontinuierlichen und nicht-konvexen gleicheckig-gleichfl¨achigen Polyeder. ¨ (German) [On the nonconvex and vertex-transitive, face-transitive polyhedra]”. In: Verh. des dritten Internat. Math.-Kongresses (1905).
7. Max Brückner. “Uber die gleicheckig-gleichfl¨achigen diskontinuierlichen und nichtkonvexen Polyeder. ¨ (German) [On the equal-angle, equal-surface, discontinuous, and non-convex polyhedra]”. In: Nova Acta Leop. (1906).
8. Max Brückner. “Zur Geschichte der Theorie der gleicheckig gleichfl¨achigen Polyeder. (German) [On the history of the theory of vertex-transitive face-transitive polyhedra]”. In: Unterrichtsblatter Math. Naturwiss. (1907).
9. Robert Webb, "Noble Faceting of Snub Cube" (2008). https://www.software3d.com/NobleSnub.php
10. Ulrich Mikloweit. “Exploring Noble Polyhedra With the Program Stella4D”. In: Bridges 2020 Conference Proceedings. Ed. by Carolyn Yackel et al. Vol. 25. 2020, pp. 257–264.
11. See under Discoverer at https://polytope.miraheze.org/wiki/List_of_noble_polyhedra
12. https://www.software3d.com/Stella.php
13. https://polytope.miraheze.org/wiki/List_of_noble_polyhedra

References 2-10 taken from [1] with the permission of the author.

Footnote.
Fn 1.  Thank-you Don Romano.


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