Emanuel Lodewijk Elte

{{short description|Dutch mathematician (1881–1943)}} '''Emanuel Lodewijk Elte''' (16 March 1881 in Amsterdam – 9 April 1943 in Sobibór)<ref name=jm>[http://www.joodsmonument.nl/person/447995/en Emanuël Lodewijk Elte] {{Webarchive|url=https://web.archive.org/web/20131213113428/http://www.joodsmonument.nl/person/447995/en |date=2013-12-13 }} at joodsmonument.nl</ref> 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.<ref name=jm/>

== 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''.<ref>{{citation | last = Elte | first = E. L. | title = The Semiregular Polytopes of the Hyperspaces | publisher = University of Groningen | location = Groningen | year = 1912 | isbn = 1-4181-7968-X | url=http://name.umdl.umich.edu/ABR2632.0001.001 }} [https://www.amazon.com/Semiregular-Polytopes-Hyperspaces-Emanuel-Lodewijk/dp/141817968X] [http://hdl.handle.net/2027/miun.abr2632.0001.001]</ref> 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.<ref>Coxeter, H.S.M. ''Regular polytopes'', 3rd Edn, Dover (1973) p. 210 (11.x Historical remarks)</ref> In the process he discovered all the main representatives of the exceptional E_''n'' family of polytopes, save only 1₄₂ which did not satisfy his definition of semiregularity.

{| class=wikitable |+ Summary of the semiregular polytopes of the first kind<ref>[http://babel.hathitrust.org/cgi/pt?seq=2;view=image;size=100;id=miun.abr2632.0001.001;u=1;num=128;page=root;orient=1 Page 128]</ref> |- !n !Elte<BR>notation !Vertices !Edges !Faces !Cells !Facets !Schläfli<BR>symbol !Coxeter<BR>symbol !Coxeter<BR>diagram |- !colspan=10| Polyhedra (Archimedean solids) |- !rowspan=10|3 || tT||12 ||18 || 4p₃+4p₆ || || |t{3,3} |||| {{CDD|node_1|3|node_1|3|node}} |- || tC||24 ||36 || 6p₈+8p₃ || || |t{4,3} |||| {{CDD|node_1|4|node_1|3|node}} |- || tO||24 ||36 || 6p₄+8p₆ || || |t{3,4} |||| {{CDD|node_1|3|node_1|4|node}} |- || tD||60 ||90 || 20p₃+12p₁₀ || || |t{5,3} |||| {{CDD|node_1|5|node_1|3|node}} |- || tI||60 ||90 || 20p₆+12p₅ || || |t{3,5} |||| {{CDD|node_1|3|node_1|5|node}} |- || TT = O||6 ||12 || (4+4)p₃ || || |r{3,3} = {3^1,1} ||0₁₁|| {{CDD|node_1|split1|nodes}} |- || CO||12 ||24 || 6p₄+8p₃ || || |r{3,4} || || {{CDD|node_1|split1-43|nodes}} |- || ID||30 ||60 || 20p₃+12p₅ || || |r{3,5} |||| {{CDD|node_1|split1-53|nodes}} |- || P_q||2q ||4q || 2p_q+qp₄ || || |t{2,q} |||| {{CDD|node_1|2x|node_1|q|node}} |- || AP_q||2q ||4q || 2p_q+2qp₃ || || |s{2,2q} |||| {{CDD|node_h|2x|node_h|2x|q|node}} |- !colspan=10| semiregular 4-polytopes |- !rowspan=12|4 || tC₅||10 ||30 || (10+20)p₃ || 5O+5T || |r{3,3,3} = {3^2,1} ||0₂₁|| {{CDD|node_1|split1|nodes|3b|nodeb}} |- || tC₈||32 ||96 || 64p₃+24p₄ || 8CO+16T || |r{4,3,3} |||| {{CDD|node_1|split1-43|nodes|3b|nodeb}} |- || tC₁₆=C₂₄(*)||48 ||96 || 96p₃ || (16+8)O || |r{3,3,4} |||| {{CDD|node_1|split1|nodes|4a|nodea}} |- || ''tC''₂₄||96 ||288 || 96''p''₃ + 144''p''₄ || 24''CO'' + 24''C'' || |r{3,4,3} |||| {{CDD|node_1|split1-43|nodes|3a|nodea}} |- || ''tC''₆₀₀||720 ||3600 || (1200 + 2400)''p''₃ || 600O + 120''I'' || |r{3,3,5} |||| {{CDD|node_1|split1|nodes|5a|nodea}} |- || ''tC''₁₂₀||1200 ||3600|| 2400''p''₃ + 720''p''₅|| 120ID+600T || |r{5,3,3} |||| {{CDD|node_1|split1-53|nodes|3b|nodeb}} |- || ''HM''₄ = C₁₆(*)||8 ||24 || 32''p''₃ ||(8+8)T || | {3,3^1,1}||1₁₁|| {{CDD|node_1|3|node|split1|nodes}} |- || – ||30 ||60 || 20''p''₃ + 20''p''₆ || (5 + 5)''tT'' || |2''t''{3,3,3} |||| {{CDD|branch_11|3ab|nodes}} |- || – ||288 ||576 || 192''p''₃ + 144''p''₈|| (24 + 24)''tC'' || |2''t''{3,4,3} || || {{CDD|label4|branch_11|3ab|nodes}} |- || – ||20 ||60 || 40''p''₃ + 30''p''₄|| 10''T'' + 20''P''₃ || |''t''_0,3{3,3,3} || || {{CDD|branch|3ab|nodes_11}} |- || – ||144 ||576 || 384''p''₃ + 288''p''₄ || 48O + 192''P''₃ || |''t''_0,3{3,4,3} || || {{CDD|label4|branch|3ab|nodes_11}} |- || – ||''q''² ||2''q''² || ''q''²''p''₄ + 2''qp''_''q'' || (''q'' + ''q'')''P''_''q'' || |2t{''q'',2,''q''} || || {{CDD|labelq|branch_10|2|branch_10|labelq}} |- !colspan=10| semiregular 5-polytopes |- !rowspan=5|5 || S₅¹ ||15 ||60 || (20+60)p₃||30T+15O ||6C₅+6tC₅ |r{3,3,3,3} = {3^3,1} ||0₃₁ || {{CDD|node_1|split1|nodes|3a|nodea|3a|nodea}} |- || S₅²||20 ||90 || 120p₃||30T+30O ||(6+6)C₅ |2r{3,3,3,3} = {3^2,2} ||0₂₂ || {{CDD|node_1|split1|nodes|3ab|nodes}} |- || HM₅||16 ||80 || 160p₃||(80+40)T ||16C₅+10C₁₆ | {3,3^2,1}||1₂₁|| {{CDD|node_1|3|node|split1|nodes|3a|nodea}} |- || Cr₅¹||40 ||240 || (80+320)p₃||160T+80O ||32tC₅+10C₁₆ |r{3,3,3,4} || || {{CDD|node_1|split1|nodes|3a|nodea|4a|nodea}} |- || Cr₅²||80 ||480 || (320+320)p₃||80T+200O ||32tC₅+10C₂₄ |2r{3,3,3,4} || || {{CDD|node_1|split1|nodes|4a3b|nodes}} |- !colspan=10| semiregular 6-polytopes |- !rowspan=5|6 || S₆¹ (*)|| || || || || |r{3⁵} = {3^4,1} || 0₄₁ || {{CDD|node_1|split1|nodes|3b|nodeb|3b|nodeb|3b|nodeb}} |- || S₆² (*)|| || || || || |2r{3⁵} = {3^3,2} || 0₃₂ || {{CDD|node_1|split1|nodes|3ab|nodes|3b|nodeb}} |- || HM₆||32 ||240 || 640p₃||(160+480)T ||32S₅+12HM₅ | {3,3^3,1} || 1₃₁ || {{CDD|node_1|3|node|split1|nodes|3a|nodea|3a|nodea}} |- || V₂₇||27 ||216 || 720p₃||1080T || 72S₅+27HM₅ | {3,3,3^2,1} || 2₂₁ || {{CDD|node_1|3|node|3|node|split1|nodes|3a|nodea}} |- || V₇₂ || 72 || 720 || 2160p₃ || 2160T || (27+27)HM₆ | {3,3^2,2} || 1₂₂ || {{CDD|node_1|3|node|split1|nodes|3ab|nodes}} |- !colspan=10|semiregular 7-polytopes |- !rowspan=7|7 || S₇¹ (*)|| || || || || |r{3⁶} = {3^5,1} || 0₅₁ || {{CDD|node_1|split1|nodes|3b|nodeb|3b|nodeb|3b|nodeb|3b|nodeb}} |- || S₇² (*)|| || || || || |2r{3⁶} = {3^4,2} || 0₄₂ || {{CDD|node_1|split1|nodes|3ab|nodes|3b|nodeb|3b|nodeb}} |- || S₇³ (*)|| || || || || |3r{3⁶} = {3^3,3} || 0₃₃ || {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes}} |- || HM₇(*) || 64 || 672|| 2240p₃ || (560+2240)T || 64S₆+14HM₆ |{3,3^4,1} || 1₄₁ || {{CDD|node_1|3|node|split1|nodes|3a|nodea|3a|nodea|3a|nodea}} |- || V₅₆ ||56 || 756 || 4032p₃||10080T || 576S₆+126Cr₆ |{3,3,3,3^2,1} || 3₂₁ ||{{CDD|node_1|3|node|3|node|3|node|split1|nodes|3a|nodea}} |- || V₁₂₆|| 126 || 2016 || 10080p₃ || 20160T || 576S₆+56V₂₇ |{3,3,3^3,1} || 2₃₁ || {{CDD|node_1|3|node|3|node|split1|nodes|3a|nodea|3a|nodea}} |- || V₅₇₆ || 576 ||10080 || 40320p₃ || (30240+20160)T || 126HM₆+56V₇₂ |{3,3^3,2} || 1₃₂ ||{{CDD|node_1|3|node|split1|nodes|3ab|nodes|3a|nodea}} |- !colspan=10|semiregular 8-polytopes |- !rowspan=6|8 || S₈¹ (*) || || || || || |r{3⁷} = {3^6,1} || 0₆₁ ||{{CDD|node_1|split1|nodes|3b|nodeb|3b|nodeb|3b|nodeb|3b|nodeb|3b|nodeb}} |- || S₈² (*) || || || || || |2r{3⁷} = {3^5,2} ||0₅₂ || {{CDD|node_1|split1|nodes|3ab|nodes|3b|nodeb|3b|nodeb|3b|nodeb}} |- || S₈³ (*)|| || || || || |3r{3⁷} = {3^4,3} || 0₄₃ ||{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3b|nodeb}} |- || HM₈(*) || 128 || 1792 || 7168p₃ ||(1792+8960)T || 128S₇+16HM₇ |{3,3^5,1} || 1₅₁ ||{{CDD|node_1|3|node|split1|nodes|3a|nodea|3a|nodea|3a|nodea|3a|nodea}} |- || V₂₁₆₀ ||2160 || 69120 || 483840p₃ || 1209600T ||17280S₇+240V₁₂₆ |{3,3,3^4,1} || 2₄₁ || {{CDD|node_1|3|node|3|node|split1|nodes|3a|nodea|3a|nodea|3a|nodea}} |- || V₂₄₀ || 240 || 6720 || 60480p₃ || 241920T || 17280S₇+2160Cr₇ |{3,3,3,3,3^2,1} || 4₂₁|| {{CDD|node_1|3|node|3|node|3|node|3|node|split1|nodes|3a|nodea}} |}

:(*) Added in this table as a sequence Elte recognized but did not enumerate explicitly

Regular dimensional families: * ''S''_''n'' = ''n''-simplex: S₃, S₄, S₅, S₆, S₇, S₈, ... * ''M''_''n'' = ''n''-cube= measure polytope: ''M''₃, ''M''₄, ''M''₅, ''M''₆, ''M''₇, ''M''₈, ... * ''HM''_''n'' = ''n''-demicube= half-measure polytope: ''HM''₃, ''HM''₄, ''M''₅, ''M''₆, ''HM''₇, ''HM''₈, ... * ''Cr''_''n'' = ''n''-orthoplex= cross polytope: ''Cr''₃, ''Cr''₄, ''Cr''₅, ''Cr''₆, ''Cr''₇, ''Cr''₈, ...

Semiregular polytopes of first order: * ''V''_''n'' = semiregular polytope with ''n'' vertices

Polygons * ''P''_''n'' = regular ''n''-gon Polyhedra: * Regular: T, C, O, I, D * Truncated: tT, tC, tO, tI, tD * Quasiregular (rectified): CO, ID * Cantellated: RCO, RID * Truncated quasiregular (omnitruncated): tCO, tID * Prismatic: P_n, AP_''n''

4-polytopes: * ''C''_''n'' = Regular 4-polytopes with ''n'' cells: C₅, C₈, C₁₆, C₂₄, C₁₂₀, C₆₀₀ * Rectified: tC₅, tC₈, tC₁₆, tC₂₄, tC₁₂₀, tC₆₀₀

== See also == * Gosset–Elte figures

==Notes== {{reflist}}

{{Sobibor extermination camp}}

{{Authority control}}

{{DEFAULTSORT:Elte, E. L.}} Category:1881 births Category:1943 deaths Category:Dutch mathematicians Category:Dutch Jews who died in the Holocaust Category:Scientists from Amsterdam Category:Dutch people who died in Sobibor extermination camp Category:Dutch civilians killed in World War II