Four-Dimensional Regular Hexagon

Fou -Dimensional Regula Hexagon by Koji Miyazaki Rbumb Topologie s uc u ale #lo, 1984 L’hexagone bgulie quad idimensionnel On comp end acilemen que les analogues quad idimensionnels du iangle Cgu- lie , du ca C e du pen agone gulie (dans ce a icle, ils son ous composks uniquemen d’a i es e ne compo en aucun ClCmen a deux dimensions) son espec i emen le abd e gulie , le cube e le dodkcabd e gulie (dans ce a icle, ils son ous compos s uniquemen de aces e ne compo en aucun ClCmen a ois dimensions). Alo s, quel polybd e es l’analogue quad idimensionnel de I’hexagone gulie , c’es -a-di e un hexagone gulie quad idimensionnel? Si nous Csol ons ce e Cnigme, nous pou ons ep ksen e un locon de neige, un nid d’abeille, un c ayon, e c. quad i-dimensionnels. P op ib C ghmb iques de I’hexagone hulie . Les p incipales p op iC b gkom iques de l’hexagone kgulie se appo an ce a icle son les sui an es: I. I1 possMe un ce cle insc i auquel ou es les a i es son adjacen es en leu milieu. 2. 11 possMe un ce cle ci consc i auquel ous les somme s ouchen . 3. Tou es les a i es son d’Cgale longueu . 4. Tous les angles in e nes son Cgaux. 5. I1 peu c ie un pa age in ini du plan. 6. Tous les somme s son si ues su les axes isomk iques idimensionnels qui se c oisen 7. C’es un pa allClogone. 8. Deux iangles gulie s mu uellemen duals peu en y i e insc i s (Figu e 2). 9. On I’ob ien pa l’in e sec ion de deux iangles gulie s qui son mu uellemen duals (Figu e 2). 10. I1 es C oi emen appa en 6 a la jux aposi ion dense de ce cles. les uns les au es su un plan a 120” (Figu e 1). Abs ac S uc u al Topology #lo, 1984 I is easily unde s ood ha he Cdimensional analogues o he egula iangle, squa e, and egula pen agon (in his pape , all a e composed o only edges and ha e no po ion o 2-space) a e he egula e ahed on, cube, and egula dode- cahed on (in his pape , all a e composed o only aces and ha e no po ion o 3-space) espec i ely. Then, which polyhed on is he Cdimensional analogue o he egula hexagon, i.e. Cdimensional egula hexagon? I his iddle is sol ed, we can see a 4-dimensional snow lake, honeycomb, pencil, e c. Geome ical P ope ies o Regula Hexagon. egula hexagon which ela e o his pape a e as ollows: The main geome ical p ope ies o he I. I has an insc ibed ci cle which all midpoin s o edges con ac . 2. I has a ci cumsc ibed ci cle which all e ices con ac . 3. All edges ha e equal leng h. 4. All in e nal angles a e equal. 5. I can essella e in ini ely a plane. 6. All e ices a e on he 3-dimensional isome ic axes which in e sec each o he in he plane a 120” (Figu e 1). 7. I is a pa allelogon. 8. I con ains wo insc ibed egula iangles which a e mu ually dual (Figu e 2). 9. I can be ob ained by he in e sec ion o wo egula iangles which a e mu ually dual (Figu e 2). 10. I closely ela es o he closes ci cle a angemen . 28 Topologie s mc u ale #lo, 1984 Figu e 1 P op i6 b gkomk iques de I’hexagone bgulie quad idimensionnel. Les p op ii s g om - iques de l’hexagone gulie men ionn es ci-dessus peu en & e anspos es a I’espace a qua e dimensions comme sui : 1. I1 possPde une sph e insc i e a laquelle son adjacen es chacune des aces de l’hexa- gone en leu milieu. 2. II possede une sphe e ci consc i e a laquelle ous les somme s ouchen . 3. Tou es les aces son iden iques. 4. Tous les angles solides son gaux. 5. I1 peu empli a I’in ini un espace B ois dimensions. 6. Tous les somme s son si ues su les axes isom iques quad idimensionnels qui se 7. C’es un pa all loed e. 8. Deux poly d es gulie s mu uellemen duals peu en y S e insc i s. 9. On I’ob ien pa l’in e sec ion de deux poly d es gulie s qui son mu uellemen duals. 10. I1 es oi emen appa en 6 a la jux aposi ion dense de sph es. c oisen les uns les au es dans un espace i ois dimensions a 109028’ (Figu e 3). Compa aison de ce ains poly6d es conside & comme des hexagones 6gulie s quad idimen- sionnels. Ce ains poly d es peu en & e consid is comme des hexagones kgulie s quad idimensionnels. Le Tableau 1 les compa e en enan comp e des p op i s ghmL iques men ionnks ci-hau . Les in e sec ions ma qu6es pa des as isques indiquen que les p op i s g om iques on C sa is ai es. Figu e 1 Geome ical P ope ies o CDimensional Regula Hexagon. The abo e men ioned geo- me ical p ope ies o he egula hexagon can be ex ended in o 4-space as ollows: 1. I has an insc ibed sphe e which all cen al poin s o aces con ac . 2. I has a ci cumsc ibed sphe e which all e ices con ac . 3. All aces a e iden ical. 4. All solid-angles a e equal. 5. I can ill in ini ely a 3-space. 6. All e ices a e on he Cdimensional isome ic axes which in e sec each o he in a 3-space a 109O28’ (Figu e 3). 7. I is a pa allelohed on. 8. I con ains wo insc ibed egula polyhed a which a e mu ually dual. 9. I can be ob ained by he in e sec ion o wo egula polyhed a which a e mu ually dual. 10. I closely ela es o he closes sphe e a angemen . Compa ison o some Polyhed a as 4Dimensional Regula Hexagon. The e a e some polyhed a which can be hough o as he Cdimensional egula hexagon. Table 1 shows hei compa ison acco ding o he abo e men ioned geome ical p ope ies. The c oss- ings ha ing as e isk show ha he geome ical p ope ies a e sa is ied. S mc u al Topology #lo, 1984 29 Figu e 3 Le cube conside 6 comme hexagone kgulie quad idimensionnel. D’ap s le Tableau 1, le cube poss de le plus g and nomb e d’as isques. Pa consequen , on peu pense que le cube es celui qui se app oche le plus de I’hexagone quad idimensionnel igulie . Feu Da id B isson a dessini des locons de neige quad idimensionnels insc i s dans des cubes comme on peu le oi B la Figu e 4. II a d’abo d comp is que les locons de neige idimensionnels son des oiles de Da id subdi is es selon la h o ie ac ale de Mandel- b o . Ensui e, il a imagin des locons quad idimensionnels p odui s pa une subdi ision similai e de la s ella oc angula de Keple . II semble que, pou hi, I’hexagone gulie quad idimensionnel ai le cube. Tou e ois, le cube ne peu pas sa is ai e aux p op i is g om iques 9 e 10, e de plus, il es de mani e ce aine I’analogue du ca . Le hombidodkaM e conside e comme hexagone bgulie quad idimensionnel. Dans ce a icle, le hombidodkca d e don ou es les aces on des diagonales de appo 1: J2 e son dual, le cuboc a d e, son consid s comme les hexagones quad idimensionnels igulie s. Ensemble, ils peu en sa is ai e ii ou es les p op i is giomi iques. En ce qui conce ne les p op i s 8 e 9, la Figu e 5 illus e commen elles son sa is ai es. De plus, ou es les a & es e les segmen s de d oi e elian le cen e du cuboc a2d e a ous ses somme s son mu uellemen igales 1 la longueu du ayon de la sphk e ci consc i e, ce qui 6 ai aussi le cas pou I’hexagone igulie . D pendan des ci cons ances, l’on adop e le hombidod cakd e ou le cuboc akd e, bien que le p emie soi sa is aisan dans la plupa des cas. 12 3 4 5 6 7 8 9 10 Cube ******** Cube Oc akd e igulie **** * Regula Oc ahed on Ti a d e onqu ** * T unca ed Te ahed on P isme hexagonal * ** * Hexagonal P ism Oc a d e onqu ** ** T unca ed Oc ahed on Cuboc a d e Rhombidod ca d e * * * * * * * Rhombic Dodecahed on ** * Cuboc ahed on Tableau 1 - Table 1 Cube as CDimensional Regula Hexagon. Acco ding o Table 1, he cube has he la ges numbe o he as e isks. The e o e, i may be hough ha he cube is closes o a Cdimensional egula hexagon. The la e Da id B isson designed 4-dimensional snow- lakes insc ibed in cubes as in Figu e 4. Fi s , he g asped ha 3-dimensional snow lakes a e he s a s o Da id subdi ided acco ding o he ac al heo y o Mandelb o . And nex , he imagined Cdimensional snow lakes such ha he Keple ’s s ella oc angula is subdi ided. I seems ha he Cdimensional egula hexagon o him was he cube. Howe e , he cube can no sa is y he geome ical p ope ies 9 and 10, and u he mo e, i is su ely he analogue o he squa e. Rhombic Dodecahed on as CDimensional Regula Hexagon. In his pape , he hombic dodecahed on all o whose aces ha e he diagonals o he a io 1: J2 and i s dual, he cuboc ahed on, a e hough o as he Cdimensional egula hexagons. They can sa is y all o he geome ical p ope ies by helping one ano he . Conce ning p ope ies 8 and 9, he si ua ions a e as in Figu e 5. And mo eo e , all o he edges and lines connec ing he body-cen e wi h all he e ices o he cuboc ahed on a e mu ually equal o he leng h o he adius o he ci cumsc ibed sphe e, as in he condi ion o he egula hexagon. I depends upon ci cums ances which o he hombic dodecahed on o cuboc ahed on is adop ed, hough he o me may be sui able in mos cases. 30 Topoiogie s uc u ale #lo, 1984 Figu e 4 Figu e 5 Flocon de neige quad idimensionnel. La Figu e 6 p sen e des cons uc ions de locons de neige quad idimensionnels a pa i de hombidod ca d es. Le 7 a il 1983, a la sugges- ion d’un japonais, la na e e spa iale Khallenge ,, a en e de ai e des locons de neige dans I’espace pou les sous ai e a I’ac ion de la g a i . Malheu eusemen , cela n’a pas ussi. Mais si cela a ai ussi, les locons de neige se aien els qu’a la Figu e 6, pa ce qu’une gou e d’eau ci culai e de ien sphe ique dans I’espace, lo squ’elle n’es pas soumise a l’ac ion de la g a i i. C’es la mZme chose dans I’espace a qua e dimensions. Nid d’abeille quad idimensionnel. La Figu e 7 p sen e un nid d’abeille quad idimen- sionnel e une abeille quad idimensionnelle. Dans un espace idimensionnel, un nid d’abeille es compose de ubes, chacun ayan la o me d’un hombidod ca d e, c’es -&di e la o me ex ieu e d’un cube a qua e dimen- sions allong le long d’un ensemble d’a i es mu uellemen pa all les. E quand ce nid d’abeille a ois dimensions es coup pa un plan pe pendiculai emen aux a i es allon- g es, le pa age gulie qui appa ai es compos uniquemen d’hexagones igulie s; en d’au es mo s, nous ape ce ons une jux aposi ion dense de ce cles. Dans ce cas, chaque hexagone gulie de ien un pa allilo-hexagone lo squ’il es p oje obliquemen su un plan. Au con ai e, la base de chaque ube de ien un hexagone gulie compos de ois losanges lo squ’il es p oje o hogonalemen su un plan pe pendiculai e aux a i es allong es. 4Dimensional Snow lake. Figu e 6 shows Cdimensional snow lakes buil acco ding o he hombic dodecahed a. On Ap il 7, 1983, he space shu le “Challenge ” a emp ed o make snow lakes in space ha ing no g a i y acco ding o a p oposal by a Japanese. Reg e ully, i ended in a ailu e. Bu , i i p o ed success ul, he snow lakes migh ha e become as in Figu e 6 because a ci cula d op o wa e would become sphe ical in space, ha ing no g a i y. So also in 4-space. 4-Dimensional Honeycomb. Figu e 7 shows a bdimensional honeycomb and a 4-dimensional bee. In 3-space, a honeycomb is composed o some ubes each o which has he shape o a hombic dodecahed on, i.e. he ou e shape o he Ccube, elonga ed along one se o mu ually pa allel edges. And when his 3-honeycomb is cu pe pendicula o he elon- ga ed edges by a plane, he egula essella ion c ea ed uses only egula hexagons; in o he wo ds, he closes ci cle a angemen appea s as he sec ion. He e, each egula hexagon becomes a pa allelo-hexagon when i is obliquely p ojec ed on o a plane. On he con a y, he base o each ube becomes a egula hexagon composed o h ee hombi when i is o hogonally p ojec ed on o a plane which is pe pendicula o he elonga ed edges. S uc u al Topology #lo, 1984 31 Donc, dans l’espace a qua e dimensions, un nid d’abeille peu i e composC de ubes ayan chacun La o me d’un hombicosa d e, c’es -a-di e La o me ex ieu e du cube B cinq dimensions allonge le long d’un ensemble d’a i es mu uellemen pa all les. E lo sque ce nid d’abeille quad idimensionnel es coupe pa un hype -plan, pe pendiculai- emen aux a & es allongees, le seau d’al Coles qui appa ai poss de uniquemen des hombidod cakd es; donc, nous ape ce ons une jux aposi ion dense de sph es. Ici, chaque hombidodeca d e de ien un pa allklo-dod caed e, pa exemple, le <<second hombidodCcaed e,, de Bilinski, lo squ’il es p oje obliquemen dans un espace B ois dimensions. Au con ai e, la base de chaque ube de ien un hombidodeca d e compos de qua e homboides lo squ’il es p oje de aGon o hogonale dans un espace ois dimensions pe pendiculai e aux a i es allongees. Une abeille quad idimensionnelle poss de une i e e un co ps, mais ois yeux, ois an ennes e ois ailes paisses comme des ca o es. The e o e, in Cspace, a honeycomb may be composed o some ubes each o which has he shape o a hombic icosahed on, i.e. he ou e shape o he Scube, elonga ed along one se o mu ually pa allel edges. And when his Choneycomb is cu pe pendicula o he elonga ed edges by a hype -plane, he <<honeycomb,, has only hombic dodecahed a; so he closes sphe e a angemen appea s as he sec ion. He e, each hombic dodecahed on becomes a pa allelododecahed on, e.g. Bilinski’s .second hombic dodecahed on*, when i is obliquely p ojec ed on o 3-space. On he con a y, he base o each ube becomes a hombic dodecahed on composed o ou homboids when i is o hogonally p ojec ed in o a 3-space which is pe pendicula o he elonga ed edges. A Cdimensional bee has one head and one body, bu h ee each o eyes, an ennae, and wings hick like ca o s. Figu e 6 S uc u al Topology #lo, 1984 33 Un c ayon quad idimensionnel. La Figu e 8 p sen e un c ayon quad idimensionnel. Dans un espace a ois dimensions, un c ayon a la o me d’un p ime hexagonal egulie don un des bou s es I’hexagone gulie ou n’impo e quel pa allilo-hexagone e don I’au e bou es aiguisien o me d’un c6ne ci culai e don la base a six hype boles comme a es. Alo s, dans un espace a qua e dimensions, un c ayon a la o me d’un hype - p isme hexagonal don un des bou s es le hombidod cakd e ou ou pa allelo- dodecaid e e don l’au e bou es aiguise en o me d’un hype -c6ne ci culai e don la base posskde douze hype boloides comme aces (des hype -a i es). &Dimensional Pencil. Figu e 8 shows a Cdimensional pencil. In 3-space, a pencil has he shape o he egula hexagonal p ism o which one end is he egula hexagon o any pa allelo-hexagon and he o he end is sha ed in he o m o a ci cula cone whose base has six hype bolas as he edges. So, in Cspace, a pencil has he shape o he hype - hexagonal p ism o which one end is he hombic dodecahed on o any pa allelo- dodecahed on and he o he end is sha ed in he o m o a hype -ci cula cone whose base has wel e hype boloids as he aces (hype -edges). Figu e 8 34 Topologie sm uc u ale #lo, 1984 Addendum. Lo s de la seconde en a i e, le le‘ sep emb e 1983, Challenge >> Cussi B ab ique des locons de neige dans l’espace, en a d’apesan eu . Bien que la d ini ion des images ob enues su un Cc an de B aun lahe A d si e (Figu e 9), il es possible de cons a e la esemblance de celles-ci a ec nos dessins pa o dina eu de locons de neige quad idimensionnels hombidod ca d iques (Figu e 10). Addendum. On Sep embe 1,1983, “Challenge ” p o ed success ul in he second a emp o make snow lakes in space ha ing no g a i y, hough he pic u es on a B aun ube we e no so clea (Figu e 9). They esemble ou compu e g aphics o hombic dodecahed a1 Mimensional snow lakes (Figu e 10). Figu e 9’ Ad esse de I’au eu : Koji Miyazaki Kobe Uni e si y 1-2-1 Tsu ukabu u Nada-Ku Kobe-Shi 657 Japan * Une g acieuse du Jou nal Asahi Figu e 10’ Add ess o he au ho : Koji Miyazaki Kobe Uni e si y 1-2-1 Tsu ukabu u Nada-Ku Kobe-Shi 657 Japan * By he cou esy o he Asahi Newspape