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