Emanuel Lodewijk Elte (16 March 1881 inAmsterdam – 9 April 1943 inSobibór)[1] was aDutchmathematician. He is noted for discovering and classifying semiregularpolytopes in dimensions four and higher.
Elte's father Hartog Elte was headmaster of a school in Amsterdam. Emanuel Elte married Rebecca Stork in 1912 in Amsterdam, when he was a teacher at a high school in that city. By 1943 the family lived inHaarlem. When on January 30 of that year a German officer was shot in that town, in reprisal a hundred inhabitants of Haarlem were transported to theCamp Vught, including Elte and his family. As Jews, he and his wife were further deported to Sobibór, where they were murdered; his two children were murdered atAuschwitz.[1]
His work rediscovered the finitesemiregular polytopes ofThorold Gosset, and further allowing not only regularfacets, but recursively also allowing one or two semiregular ones. These were enumerated in his 1912 book,The Semiregular Polytopes of the Hyperspaces.[2] He called themsemiregular polytopes of the first kind, limiting his search to one or two types of regular or semiregulark-faces. These polytopes and more were rediscovered again byCoxeter, and renamed as a part of a larger class ofuniform polytopes.[3] In the process he discovered all the main representatives of the exceptional En family of polytopes, save only142 which did not satisfy his definition of semiregularity.
| n | Elte notation | Vertices | Edges | Faces | Cells | Facets | Schläfli symbol | Coxeter symbol | Coxeter diagram |
|---|---|---|---|---|---|---|---|---|---|
| Polyhedra (Archimedean solids) | |||||||||
| 3 | tT | 12 | 18 | 4p3+4p6 | t{3,3} |    | |||
| tC | 24 | 36 | 6p8+8p3 | t{4,3} |  | ||||
| tO | 24 | 36 | 6p4+8p6 | t{3,4} | | ||||
| tD | 60 | 90 | 20p3+12p10 | t{5,3} |  | ||||
| tI | 60 | 90 | 20p6+12p5 | t{3,5} | | ||||
| TT = O | 6 | 12 | (4+4)p3 | r{3,3} = {31,1} | 011 |   | |||
| CO | 12 | 24 | 6p4+8p3 | r{3,4} |  | ||||
| ID | 30 | 60 | 20p3+12p5 | r{3,5} |  | ||||
| Pq | 2q | 4q | 2pq+qp4 | t{2,q} |   | ||||
| APq | 2q | 4q | 2pq+2qp3 | s{2,2q} |  | ||||
| semiregular4-polytopes | |||||||||
| 4 | tC5 | 10 | 30 | (10+20)p3 | 5O+5T | r{3,3,3} = {32,1} | 021 |   | |
| tC8 | 32 | 96 | 64p3+24p4 | 8CO+16T | r{4,3,3} | | |||
| tC16=C24(*) | 48 | 96 | 96p3 | (16+8)O | r{3,3,4} |   | |||
| tC24 | 96 | 288 | 96p3 + 144p4 | 24CO + 24C | r{3,4,3} |  | |||
| tC600 | 720 | 3600 | (1200 + 2400)p3 | 600O + 120I | r{3,3,5} |  | |||
| tC120 | 1200 | 3600 | 2400p3 + 720p5 | 120ID+600T | r{5,3,3} | | |||
| HM4 = C16(*) | 8 | 24 | 32p3 | (8+8)T | {3,31,1} | 111 | | ||
| – | 30 | 60 | 20p3 + 20p6 | (5 + 5)tT | 2t{3,3,3} |   | |||
| – | 288 | 576 | 192p3 + 144p8 | (24 + 24)tC | 2t{3,4,3} |  | |||
| – | 20 | 60 | 40p3 + 30p4 | 10T + 20P3 | t0,3{3,3,3} |   | |||
| – | 144 | 576 | 384p3 + 288p4 | 48O + 192P3 | t0,3{3,4,3} | | |||
| – | q2 | 2q2 | q2p4 + 2qpq | (q +q)Pq | 2t{q,2,q} |    | |||
| semiregular5-polytopes | |||||||||
| 5 | S51 | 15 | 60 | (20+60)p3 | 30T+15O | 6C5+6tC5 | r{3,3,3,3} = {33,1} | 031 | |
| S52 | 20 | 90 | 120p3 | 30T+30O | (6+6)C5 | 2r{3,3,3,3} = {32,2} | 022 | | |
| HM5 | 16 | 80 | 160p3 | (80+40)T | 16C5+10C16 | {3,32,1} | 121 | | |
| Cr51 | 40 | 240 | (80+320)p3 | 160T+80O | 32tC5+10C16 | r{3,3,3,4} | | ||
| Cr52 | 80 | 480 | (320+320)p3 | 80T+200O | 32tC5+10C24 | 2r{3,3,3,4} |  | ||
| semiregular6-polytopes | |||||||||
| 6 | S61 (*) | r{35} = {34,1} | 041 | | |||||
| S62 (*) | 2r{35} = {33,2} | 032 | | ||||||
| HM6 | 32 | 240 | 640p3 | (160+480)T | 32S5+12HM5 | {3,33,1} | 131 | | |
| V27 | 27 | 216 | 720p3 | 1080T | 72S5+27HM5 | {3,3,32,1} | 221 | | |
| V72 | 72 | 720 | 2160p3 | 2160T | (27+27)HM6 | {3,32,2} | 122 | | |
| semiregular7-polytopes | |||||||||
| 7 | S71 (*) | r{36} = {35,1} | 051 | | |||||
| S72 (*) | 2r{36} = {34,2} | 042 | | ||||||
| S73 (*) | 3r{36} = {33,3} | 033 | | ||||||
| HM7(*) | 64 | 672 | 2240p3 | (560+2240)T | 64S6+14HM6 | {3,34,1} | 141 | | |
| V56 | 56 | 756 | 4032p3 | 10080T | 576S6+126Cr6 | {3,3,3,32,1} | 321 | | |
| V126 | 126 | 2016 | 10080p3 | 20160T | 576S6+56V27 | {3,3,33,1} | 231 | | |
| V576 | 576 | 10080 | 40320p3 | (30240+20160)T | 126HM6+56V72 | {3,33,2} | 132 | | |
| semiregular8-polytopes | |||||||||
| 8 | S81 (*) | r{37} = {36,1} | 061 | | |||||
| S82 (*) | 2r{37} = {35,2} | 052 | | ||||||
| S83 (*) | 3r{37} = {34,3} | 043 | | ||||||
| HM8(*) | 128 | 1792 | 7168p3 | (1792+8960)T | 128S7+16HM7 | {3,35,1} | 151 | | |
| V2160 | 2160 | 69120 | 483840p3 | 1209600T | 17280S7+240V126 | {3,3,34,1} | 241 | | |
| V240 | 240 | 6720 | 60480p3 | 241920T | 17280S7+2160Cr7 | {3,3,3,3,32,1} | 421 | | |
Regular dimensional families:
Semiregular polytopes of first order:
Polygons
Polyhedra:
4-polytopes:
| International | |
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| National | |
| Academics | |
