Timeline

Icosahedral die from Egypt, 2nd century B.C. to 4th century A.D.

This is a (partially complete) list of important events in the study of polytopes.

Before year 1 CE

• The Sumerians were using tessellations in their art as early as the fourth millennium BCE.
• The Platonic solids are thought to have been discovered sometime in the first millennium BCE.
• The Archimedean solids were attributed in later manuscripts to the third-century-BCE mathematician Archimedes.

Small and great stellated dodecahedra in a 1619 book by Kepler

1 - 1800

• Artwork depicting the Kepler-Poinsot solids can be found as far back as the 15th century.
  • Around 1619, Johannes Kepler recognized the small and great stellated dodecahedron as regular.
  • He also coined names to all the convex uniform polyhedra, aka Archimedean polyhedra.
• Kepler also wrote on tessellations in 1619.

Views of the tesseract in Hinton's 1904 book

1800 - 1900

• In 1809, Louis Poinsot recognized the great dodecahedron and great icosahedron as regular.
  • Three years later, the list of nonconvex finite regular ("star") polyhedra was proved complete by Augustin Cauchy.

• Ludwig Schläfli first described the convex regular polychora in the mid-19th century, as well as some of the nonconvex finite ones.
  • Edmund Hess published a list of all of the nonconvex finite regular polychora in 1883. These became known as the Schläfli-Hess polychora.

• As early as 1888, Charles Howard Hinton coined the term "tesseract".

• In 1891, crystallographer Evgraf Fedorov proved that there are seventeen symmetries of the Euclidean plane, sometimes known as "wallpaper groups". His work would begin the formal mathematical study of tessellations.

• Alicia Boole Stott coined the term "polytope" in the latter half of the 19th century.

• Thorold Gosset discovered the semiregular polytopes 2₂₁, 3₂₁, and 4₂₁ in 1896-1897.

Some of Bruckner's models

Wenninger and a student

1900 - 1990

• In 1905, the Poincaré disk model of the hyperbolic plane became widely known, although it had been discussed for decades beforehand.

• Max Bruckner published his collections of noble polytopes in 1906.

• Alicia Boole Stott in 1910 published about what became known as "Stott expansions".

• Harold Scott MacDonald Coxeter recieved his Ph.D. in 1931.
  • In 1926, Coxeter and John Flinders Petrie discovered the regular skew apeirohedra: the mucube, muoctahedron, and mutetrahedron.
  • In 1938, he, Petrie, and others published The Fifty-Nine Icosahedra, a list of the stellations of the icosahedron.
  • Coxeter, J. C. P. Miller, and others published a list of uniform polyhedra in 1954.
    • John Skilling proved the list complete in 1975.
  • In 1974, Coxeter published Regular Complex Polytopes.
  • Coxeter contributed to the development of Coxeter-Dynkin diagrams, was friends with M. C. Escher, and inspired the work of Buckminster Fuller.

• Branko Grünbaum received his Ph.D. in 1957. He would author hundreds of papers on discrete geometry and abstract polytopes over the next fifty years.

• In 1966, Norman Johnson received his Ph.D. under the supervision of H.S.M. Coxeter.
  • In this year, he also published a list of non-uniform convex regular-faced polyhedra, which came to be known as the Johnson solids.
    • Victor Zalgaller proved the list to be complete in 1969.
  • Johnson also gave names to all of the nonconvex uniform polyhedra.

• In 1970, Bonnie Stewart published Adventures among the Toroids, in which he and Norman Johnson defined and began to enumerate the Quasi-convex Stewart toroids.

• In 1971, Father Magnus Wenninger published Polyhedron Models, the first time that pictures of the uniform polyhedra had been widely published.

• In 1976, Branko Grünbaum discovers the 11-cell. Motivated by this discovery of a combinatorial polytope, he creates a definition for polystromata, a precursor to the modern definition of abstract polytopes.

• In 1977, Branko Grünbaum extends the definition of skew polyhedron to include polyhedra with skew faces, and enumerates 27 new regular polyhedra:
  • The Petrials of the already known polyhedra (12 polyhedra)
  • The blended polyhedra (12 polyhedra)
  • The trihelical square tiling (1 polyhedron)
  • The tetrahelical triangular tiling (1 polyhedron)
  • The skewed muoctahedron (1 polyhedron)

• In the 1980s, Gerd and Roswitha Blind listed the Blind polytopes, a subset of convex regular-faced or "CRF" polytopes.

• In 1981, Nicolaas Govert de Bruijn published an algebraic theory of the Penrose tiling.

• In 1985, Andreas Dress proved that there were exactly 48 regular polyhedra by Grünbaum's definition which includes skew polyhedra.

A cross-section of rapsady, rendered with POV-Ray.

A projection of siggissido with its great icosahedron cells visible, rendered in Stella4D.

1990 - present

• In 1990, Jonathan Bowers' polychoron search began.
  • In 1993 he found rapsady.
  • In 1997 he found the idcossid, dircospid, sidtap, and gidtap regiments.
  • In 1998 he discovered the sabbadipady regiment.
  • Other polytopists contributed as well. In 1999, George Olshevsky came up with the idea of swirlprisms or swirlchora, which display discretized Hopf fibrations.
  • In 2001-2, Iquipadah was found.

• In 1996, George Hart began publishing his work on polyhedron models.

• In 2000, Richard Klitzing published a list of convex segmentochora.
  • It already contained the truncated tetrahedral alterprism, the very first example of a scaliform polychoron.
  • In 2010, he provided a deeper and unifying insight into snubbing, holosnub, and more general alternations.

• In 2004, Alex Doskey discovered four Quasi-convex Stewart toroids made by dissecting Archimedean solids and expanding the parts with prisms. These were the first quasi-convex Stewart toroids whose hulls were not regular faced.

• In 2005, Wendy Krieger released a paper Walls and Bridges, which introduced prism, tegum, pyramid, and comb products.

• By the year 2006, the number of known uniform polychora had risen to 1849, due to the work of Bowers, Olshevsky, Mason Green, Hironori Sakamoto, and others.

• In 2007, Robert Webb released the Stella software, which is extremely useful for exploring uniform polychora and their scaliform relatives.

• In 2008, Mathieu Dutour Sikirić and Wendy Myrvold determined the number of Blind polytopes created by diminishing the hexacosichoron.

• In 2014, Higherspace forum users discovered many CRF polychora that unprecedentedly used bilunabirotundae and triangular hebesphenorotundae as cells.

• In April 2019, Robert Webb discovered the Webb toroid, which could be combined with the holey monster to create a quasi-convex Stewart toroid of genus 87, breaking the record held by the holey monster since the publication of Adventures among the Toroids in 1964.

• In August 2019, Klitzing brought certain compounds of hexacosichora including Pedisna to Bowers' attention, leading to a large wave of discoveries.

• In late 2020 and early 2021, Polytope Discord user _Geometer made discoveries that lead to the discovery of hundreds of new uniform polychora.
  • In September 2020, sidditsphit and gidditsphit were found, the first uniform polychora since 2006.
  • In early October, setut and getut were found.
  • In late October, pecuexdap and pecuexidfap were found.
  • In January 2021, several hundred uniform polychora in the disdi regiment were found, the first being siggissido.
  • In October 2021, tesapdid was found by accident.

• In 2022, the addition of a faceting feature in Miratope led to an even larger discovery wave of scaliforms and fissary uniforms including Dexdap. Miratope also allowed many 6D regiments to be counted.
  • _Geometer found the infinite family of uniform polygonal duoprismatic spinoalterprisms during this time.

External links

• Starck, Maurice. "Polyhedra's history".