Английская Википедия:Emanuel Lodewijk Elte

Материал из Онлайн справочника
Перейти к навигацииПерейти к поиску

Шаблон:Short description Emanuel Lodewijk Elte (16 March 1881 in Amsterdam – 9 April 1943 in Sobibór)[1] was a Dutch mathematician. He is noted for discovering and classifying semiregular polytopes 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 in Haarlem. 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 the Camp 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 at Auschwitz.[1]

Elte's semiregular polytopes of the first kind

His work rediscovered the finite semiregular polytopes of Thorold Gosset, and further allowing not only regular facets, 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 them semiregular polytopes of the first kind, limiting his search to one or two types of regular or semiregular k-faces. These polytopes and more were rediscovered again by Coxeter, and renamed as a part of a larger class of uniform polytopes.[3] In the process he discovered all the main representatives of the exceptional En family of polytopes, save only 142 which did not satisfy his definition of semiregularity.

Summary of the semiregular polytopes of the first kind[4]
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} Шаблон:CDD
tC 24 36 6p8+8p3 t{4,3} Шаблон:CDD
tO 24 36 6p4+8p6 t{3,4} Шаблон:CDD
tD 60 90 20p3+12p10 t{5,3} Шаблон:CDD
tI 60 90 20p6+12p5 t{3,5} Шаблон:CDD
TT = O 6 12 (4+4)p3 r{3,3} = {31,1} 011 Шаблон:CDD
CO 12 24 6p4+8p3 r{3,4} Шаблон:CDD
ID 30 60 20p3+12p5 r{3,5} Шаблон:CDD
Pq 2q 4q 2pq+qp4 t{2,q} Шаблон:CDD
APq 2q 4q 2pq+2qp3 s{2,2q} Шаблон:CDD
semiregular 4-polytopes
4 tC5 10 30 (10+20)p3 5O+5T r{3,3,3} = {32,1} 021 Шаблон:CDD
tC8 32 96 64p3+24p4 8CO+16T r{4,3,3} Шаблон:CDD
tC16=C24(*) 48 96 96p3 (16+8)O r{3,3,4} Шаблон:CDD
tC24 96 288 96p3 + 144p4 24CO + 24C r{3,4,3} Шаблон:CDD
tC600 720 3600 (1200 + 2400)p3 600O + 120I r{3,3,5} Шаблон:CDD
tC120 1200 3600 2400p3 + 720p5 120ID+600T r{5,3,3} Шаблон:CDD
HM4 = C16(*) 8 24 32p3 (8+8)T {3,31,1} 111 Шаблон:CDD
30 60 20p3 + 20p6 (5 + 5)tT 2t{3,3,3} Шаблон:CDD
288 576 192p3 + 144p8 (24 + 24)tC 2t{3,4,3} Шаблон:CDD
20 60 40p3 + 30p4 10T + 20P3 t0,3{3,3,3} Шаблон:CDD
144 576 384p3 + 288p4 48O + 192P3 t0,3{3,4,3} Шаблон:CDD
q2 2q2 q2p4 + 2qpq (q + q)Pq 2t{q,2,q} Шаблон:CDD
semiregular 5-polytopes
5 S51 15 60 (20+60)p3 30T+15O 6C5+6tC5 r{3,3,3,3} = {33,1} 031 Шаблон:CDD
S52 20 90 120p3 30T+30O (6+6)C5 2r{3,3,3,3} = {32,2} 022 Шаблон:CDD
HM5 16 80 160p3 (80+40)T 16C5+10C16 {3,32,1} 121 Шаблон:CDD
Cr51 40 240 (80+320)p3 160T+80O 32tC5+10C16 r{3,3,3,4} Шаблон:CDD
Cr52 80 480 (320+320)p3 80T+200O 32tC5+10C24 2r{3,3,3,4} Шаблон:CDD
semiregular 6-polytopes
6 S61 (*) r{35} = {34,1} 041 Шаблон:CDD
S62 (*) 2r{35} = {33,2} 032 Шаблон:CDD
HM6 32 240 640p3 (160+480)T 32S5+12HM5 {3,33,1} 131 Шаблон:CDD
V27 27 216 720p3 1080T 72S5+27HM5 {3,3,32,1} 221 Шаблон:CDD
V72 72 720 2160p3 2160T (27+27)HM6 {3,32,2} 122 Шаблон:CDD
semiregular 7-polytopes
7 S71 (*) r{36} = {35,1} 051 Шаблон:CDD
S72 (*) 2r{36} = {34,2} 042 Шаблон:CDD
S73 (*) 3r{36} = {33,3} 033 Шаблон:CDD
HM7(*) 64 672 2240p3 (560+2240)T 64S6+14HM6 {3,34,1} 141 Шаблон:CDD
V56 56 756 4032p3 10080T 576S6+126Cr6 {3,3,3,32,1} 321 Шаблон:CDD
V126 126 2016 10080p3 20160T 576S6+56V27 {3,3,33,1} 231 Шаблон:CDD
V576 576 10080 40320p3 (30240+20160)T 126HM6+56V72 {3,33,2} 132 Шаблон:CDD
semiregular 8-polytopes
8 S81 (*) r{37} = {36,1} 061 Шаблон:CDD
S82 (*) 2r{37} = {35,2} 052 Шаблон:CDD
S83 (*) 3r{37} = {34,3} 043 Шаблон:CDD
HM8(*) 128 1792 7168p3 (1792+8960)T 128S7+16HM7 {3,35,1} 151 Шаблон:CDD
V2160 2160 69120 483840p3 1209600T 17280S7+240V126 {3,3,34,1} 241 Шаблон:CDD
V240 240 6720 60480p3 241920T 17280S7+2160Cr7 {3,3,3,3,32,1} 421 Шаблон:CDD
(*) Added in this table as a sequence Elte recognized but did not enumerate explicitly

Regular dimensional families:

Semiregular polytopes of first order:

  • Vn = semiregular polytope with n vertices

Polygons

Polyhedra:

4-polytopes:

See also

Notes

Шаблон:Reflist

Шаблон:Sobibor extermination camp

Шаблон:Authority control

  1. 1,0 1,1 Emanuël Lodewijk Elte at joodsmonument.nl
  2. Шаблон:Citation [1] [2]
  3. Coxeter, H.S.M. Regular polytopes, 3rd Edn, Dover (1973) p. 210 (11.x Historical remarks)
  4. Page 128