Isoperimetric Problem for Polyhedra
(best polyhedral approximation of the sphere)
Lhuilier 1782, Steiner 1842, Lindelöf 1869, Steinitz 1927,
Goldberg 1933, Fejes Tóth 1948, Pólya 1954
Polyhedron of greatest volume V of a given number of faces and a
given surface area S?
Polyhedron of least volume circumscribed about a sphere?
V2
Maximize Isoperimetric Quotient for solids: IQ = 36π 3 ≤ 1
S
(with equality only for the sphere)
Polyhedron
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron
IQ(P )
π
√
Upper Bound
π
√
≃ 0.302
6 3
π
6 ≃ 0.524
π
√
≃ 0.605
3 3
πτ 7/2
≃ 0.755
3·55/4
4
πτ√
≃ 0.829
15 3
6 3
π
6
≃ 0.637
πτ 7/2
3·55/4
≃ 0.851
IQ of Platonic solids
(τ =
√
1+ 5
:
2
golden mean)
Platonic Solids
Conjecture [Steiner 1842]
Each of the 5 Platonic solids is the best of all isomorphic polyhedra.
(still open for the icosahedron)
Medial: simple polyhedron which all facets are ⌊6 −
24
⌋-gons.
⌊7 − n+4
24
n+4 ⌋-gons
or
F36
IQ(Icosahedron) < IQ(F36) ≃ 0.848
Conjecture [Goldberg 1933]
The polyhedron with m (6= 11, 13) facets and with greatest IQ is medial.
Polyhedron
IQ(P )
Dodecahedron
≃ 0.755
Truncated icosahedron ≃ 0.9058
Chamfered dodecahedron ≃ 0.928
Sphere
1
Upper Bound
πτ 7/2
3·55/4
≃ 0.755
≃ 0.9065
≃ 0.929
1
Conjecture [Grace 1963]
The polyhedron with n (6= 18, 22) vertices lying on the unit sphere with
highest volume is dual medial.
Goldberg Medial Polyhedra
Fullerenes
A fullerene Fn is a simple polyhedron (carbon molecule) which
n vertices - the carbons atoms - are arranged in 12 pentagons and
( n2−10) hexagons. The 32 n edges correspond to carbon-carbon bonds.
⇒ for n ≥ 12
Fullerene = Medial
• Fn can be constructed for all even n ≥ 20 except n = 22.
• 1, 2, 3, . . . , 1812 isomers Fn for n = 20, 28, 30, . . . , 60
• preferable Cn satisfies isolated pentagons rule
• C80(Ih) unique preferable fullerene with symmetry Ih for n = 80
buckminsterfullerene C60(Ih)
soccer ball
F36(D6h)
Icosahedral Fullerenes
Call icosahedral any fullerenes with symmetry group Ih or I.
• All icosahedral fullerenes are preferable, except F20(Ih)
• n = 20T where T = a2 + ab + b2
• Ih for a = b 6= 0 or ab = 0 (extended icosahedral group),
I otherwise (proper icosahedral group).
C60 = (1, 1)-dodecahedron
Buckminsterfullerene - truncated icosahedron
C80 = (2, 0)-dodecahedron
Goldberg chamfered dodecahedron (1933)
Icosadeltahedra
∗
Call icosadeltahedra any dual icosahedral fullerenes C20T
(Ih)
∗
or C20T
(I)
• Geodesic dome [Fuller]
• Capsids of viruses [Caspar and Klug]
O
45
4590
3459
1450
1237
1267
2378
15
34
3489
1560
12
23
1256
2348
O
34
15
1560
45
3489
1256
2348
3459
1450
4590
1267
2378
1237
12
23
Dual C60(Ih)
Pentakisdodecahedron {3, 5+}
Graviation [1952]
∗
(Ih)
omnicapped dodecahedron C60
Triangulation Number
T = a2 + b2 + ab
∗
(Ih )
C80
∗
(I)
C140
∗
(Ih) as omnicapped buckminsterfullerene
C180
Geodesic Domes
(a, b)
Fullerene
Geodesic dome
∗
(1, 0)
F20
(Ih)
One of Salvatore Dali houses
∗
(1, 1)
C60
(Ih)
Arctic Institute, Baffin Island
∗
(2, 0)
C80
(Ih)
Playground toy, Kjarvalstadir, Iceland
∗
(3, 0)
C180
(Ih)
Bachelor officers quarters, US Air Force, Korea
∗
(2, 2)
C240
(Ih)
U.S.S. Leyte
(3, 1) C260(I)laevo
Statue in Gagarin city, Russia
∗
(4, 0)
C320
(Ih)
Geodesic Sphere, Mt. Washington, New Hampshire
∗
(5, 0)
C500
(Ih)
US pavilion, Trade Fair 1956, Kabul, Afghanistan
(3, 3)
C540(Ih)
Hafnarfjördur dome, Iceland
∗
(Ih)
(6, 0)
C720
Radome, Arctic DEW
(6, 0)
C720(Ih)
Bath at a golf course, Tokyo
∗
(4, 4)
C960
(Ih)
Sky Eye radio telescope
∗
(8, 0)
C1280
(Ih ) German pavilion, Osaka Expo and Accra dome, Ghana
(6, 6)
C2160(Ih )
Children camp pavilion, Vyshod, Russia
(12, 0)
C2880(Ih )
Children camp pavilion, Kirov, Russia
∗
(8, 8)
C3840
(Ih )
Spaceship Earth, Epcot center, Florida
∗
(16, 0)
C5120
(Ih )
Test planetarium, Iena 1922
(16, 0)
C5120(Ih )
Internal layer of US pavilion, Expo 1967, Montreal
∗
(18, 0)
C6480
(Ih )
Géode du Musée des Sciences, La Villette, Paris
(18, 18) C19440(Ih)
Union Tank Car Co., Baton Rouge, Louisiana
∗
Icosadeltahedron C180
(Ih)
Bachelor officers quarters, US Air Force, Korea
Capsids of Viruses
(a, b)
Fullerene
Icosahedral virus capsid
∗
(1, 0)
F20
(Ih)
Gemini virus
∗
(1, 1)
C60
(Ih)
Turnip yellow mosaic virus
∗
(2, 0)
C80
(Ih)
Bacteriophage ΦR
∗
(2, 1) C140
(I)laevo
Rabbit papilloma virus
∗
(1, 2) C140
(I)dextro
Human wart virus
(3, 1) C260(I)laevo
Rotavirus
∗
(4, 0)
C320
(Ih)
Herpes virus, varicella
∗
(5, 0)
C500
(Ih)
Infectious canine hepatitis virus, adenovirus
∗
(6, 0)
C720
(Ih)
HTLV-1
∗
(I)laevo
(6, 3) C1260
HIV-1, usually admitted value: (6, 3)
∗
(7, 7) C2940
(Ih)
Iridovirus, also considered value: (10, 4)
∗
Icosadeltahedron C720
(Ih)
the icosahedral structure of the HTLV-1
(a, 0)-Dodecahedron
The (a, 0)-dodecahedron is the icosahedral fullerene C20T (Ih) with
T = a2. For a = 1, 2 and 3: regular dodecahedron, Goldberg chamfered dodecahedron and polyhedron from Sanpo-Kiriko-Shu.
(2, 0)-dodecahedron C80(Ih)
Goldberg chamfered dodecahedron (1933)
(3, 0)-dodecahedron C180(Ih)
from Sanpo-Kiriko-Shu, Edo era
Proposition
Besides the dodecahedron, the icosahedron and the chamfered dodecahedron, the (a, 0)-dodecahedron C20a2 (Ih) and its dual are not
ℓ1-embeddable
∗
Computer simulated adenovirus C500
(Ih) with its spikes
Triangulations, Spherical Wavelets
Dual 4-chamfered cube ⋄∗2048
∗
(Ih)
Dual 4-chamfered dodecahedron C5120
(US Pavilion, Expo ’67, Montreal)
Triangulations, Spherical Wavelets
Hamiltonian circuit on the dual 4-chamfered cube ⋄∗2048
Onion-Like Metallic Clusters
Palladium icosahedral 5-cluster P d561L60(OAc)180
α
Outer shell
∗
1
C20
(Ih)
2 RhomDode∗80(Oh)
4 RhomDode∗320(Oh)
∗
(Ih)
5
C500
Total # of atoms
Metallic cluster
13
55
309
561
[Au13(P M e2P h)10Cl2]3+
Au55(P P h3)12Cl6
P t309(P hen36O30±10 )
P d561L60(OAc)180
Icosahedral and cuboctahedral metallic clusters