_{1}

^{*}

The architecture of the Great Pyramid at Giza is based on fascinating golden mean geometry. Recently the ratio of the in-sphere volume to the pyramid volume was calculated. One yields as result
*R*
_{V} = π
⋅
*φ*^{5}, where
is the golden mean. It is important that the number
*φ*
^{5} is a fundamental constant of nature describing phase transition from microscopic to cosmic scale. In this contribution the relatively small volume ratio of the Great Pyramid was compared to that of selected convex polyhedral solids such as the
*Platonic *solids respectively the face-rich truncated icosahedron (bucky ball) as one of
*Archimedes*’ solids leading to effective filling of the polyhedron by its in-sphere and therefore the highest volume ratio of the selected examples. The smallest ratio was found for the Great Pyramid. A regression analysis delivers the highly reliable volume ratio relation
, where
*nF* represents the number of polyhedron faces and b approximates the silver mean. For less-symmetrical solids with a unique axis (tetragonal pyramids) the in-sphere can be replaced by a biaxial ellipsoid of maximum volume to adjust the
*R*
_{V} relation more reliably.

Recently the golden mean architecture of the Great Pyramid at Giza was investigated with a surprising result [

φ = 5 − 1 2 = 0.6180339887 is the golden mean. The same result can be obtained when calculating the respective areas ratio. This connection is valid for all regular

convex polyhedral solids. Unfortunately, in the appendix of reference [

In this contribution the in-sphere to polyhedron volume ratios for selected convex solids such as Platonic solids [

Known elementary techniques of Euclidian coordinate geometry in ℝ 3 were used to determine area, volume, in-sphere volume and the in-sphere to polyhedron volume ratio for selected convex polyhedra, using the universal number of the golden mean when dealing with polyhedra of icosahedral symmetry. In contrast to irregular solids [

For a convex polyhedron the radius of its in-sphere r_{i} is related to the ratio between its volume V_{p} and its surface area A_{p} by

r i = 3 V p A p (1)

It could not be found out who actually first derived this relationship. However, in the Appendix an explanation for the more general application of the relation is given as the weighted mean “in-sphere” half-distance when the polyhedral void can’t be adapted by a simple sphere, exemplified by the case of the truncated octahedron.

One can derive for the volume ratio using relation (1)

V s p h V p = 36 π V p 2 A p 3 (2)

and further for the surface area ratio

A s p h A p = 36 π V p 2 A p 3 (3)

This proves that volume ratio and surface area ratio are equal

V s p h V p = A s p h A p (4)

In the following the mentioned ratios were calculated for selected convex solids. The polyhedron notation is given by the symbol [ p i F i ] , where p is the polygon multiplicity and F is the number of faces. The number V of vertices can be derived by means of the following equation [

V = 1 2 ∑ p i F i − ∑ F i + 2 (5)

using the Euler relation for convex polyhedra [

F + V − E = 2 (6)

where E = 1 2 ∑ p i F i means the number of edges. Don’t confuse V with symbols of

A list of notations used in the following was summarized in

We begin with the truncated icosahedron, a convex isogonal non-prismatic solid of symmetry group I_{h} with 32 faces, known from the soccer ball structure of C_{60}, where 60 C atoms are placed at the 60 vertices. The symmetry group tells us that the golden mean must be involved in the formulas for the polyhedron volume respectively area.

Term | Symbol |
---|---|

Polyhedral volume | V_{p} |

In-sphere volume | V_{sph} |

In-ellipsoid volume | V_{el } |

Polyhedral area | A_{p} |

In-sphere area | A_{sph} |

In-sphere radius | r_{i} |

In-ellipsoid radii | r_{x}, r_{z} |

Polyhedron edge | a |

Polyhedron height | h |

Golden mean | φ |

Big phi | ϕ = φ + 1 |

Truncated icosahedron (soccer ball) [5^{12}6^{20}]:

V p = ( 21 φ − 2 + φ 2 ) a 3 = 55.2877308 ⋅ a 3 (7)

A p = ( 20 ⋅ 3 2 3 + 12 5 4 ⋅ φ 3 5 ) a 2 = 72.607253 ⋅ a 2 (8)

r i = 3 V p A p = 21 φ − 2 + φ 2 5 ( 2 3 + ( φ 3 5 ) − 1 2 ) a = 2.284388754 ⋅ a ≈ 2 φ − 1 a (9)

V s p h = π ⋅ 6 2 5 3 ( 7 φ − 2 + φ 6 2 3 + ( φ 3 5 ) − 1 2 ) 3 a 3 = π ⋅ 15.89456977 ⋅ a 3 ≈ π ⋅ ( 4 3 ) 5 / 4 ⋅ φ − 5 a 3 (10)

V s p h V p = π ⋅ 36 V p 2 A p 3 = π ⋅ 0.28748819 = 0.903170795 (11)

A s p h = 4 π r i 3 a 2 = π ⋅ 36 ( V p A p ) 2 a 2 (12)

A s p h A p = π ⋅ 36 V p 2 A p 3 (13)

Triacontahedron [4^{30}] (Kepler zonohedron K_{30} see Appendix):

V p = 4 5 + 2 5 a 3 (14)

A p = 12 5 a 2 = 26.8328157 ⋅ a 2 (15)

r i = 5 + 2 5 5 a = 1.37638192 ⋅ a (16)

V s p h V p = 4 3 π r i 3 = π ⋅ φ − 3 15 = 0.8872000 (17)

A s p h = 4 π ( 1 + 2 5 ) a 2 = 23.806074 ⋅ a 2 (18)

A s p h A p = 4 π r i 2 = π ⋅ φ − 3 15 = 0.8872000 (19)

Hexakishexahedron [3^{24}]:

V p = 3 2 a 3 (20)

r i = 3 2 5 a (21)

V s p h = π ⋅ 9 10 5 a 3 = 1.2644666 ⋅ a 3 (22)

V s p h V p = π ⋅ 3 5 5 = 0.842977767 (23)

A p = 3 5 a 2 (24)

A s p h = π ⋅ 9 5 a 2 = 5.65486677 ⋅ a 2 (25)

A s p h A p = π ⋅ 3 5 5 = π ⋅ 3 φ − 5 + φ 5 = 0.842977767 (26)

Icosahedron [3^{20}]:

V p = 5 6 φ − 2 a 3 (27)

r i = φ − 2 2 3 a (28)

V s p h = π ⋅ φ − 6 18 3 a 3 (29)

V s p h V p = π ⋅ φ − 4 15 3 = π ⋅ 0.263814507 = 0.8287977 (30)

A p = 5 ⋅ 3 a 2 (31)

A s p h = π φ − 4 3 a 2 (32)

A s p h A p = π ⋅ φ − 4 15 3 (33)

Truncated Octahedron [4^{6}6^{8}] (α-cage of the sodalite structure): (Combination of the octahedron with its dual polyhedron)

Like the cube this polyhedron is space-filling. However, the distances from the center to the different polygon faces are also different, and we apply a weighted in-sphere radius as given by equation one. The reason for this approximation is explained in the Appendix.

V p = 8 2 a 3 = 11.3137085 ⋅ a 3 (34)

r i = 4 2 1 + 2 3 a = 1.2671876 ⋅ a (35)

V s p h = 4 3 π ⋅ ( 4 2 1 + 2 3 ) 3 ⋅ a 3 = 8.523370381 ⋅ a 3 (36)

V s p h V p = 0.2398040 ⋅ π = 0.753366624 (37)

A p = 6 ( 1 + 2 3 ) a 2 = 26.7846097 ⋅ a 2 (38)

A s p h = 4 π ( 4 2 1 + 2 3 ) 2 ⋅ a 2 = 6.42305774 ⋅ π ⋅ a 2 (39)

A s p h A p = π 3 ( 4 1 + 2 3 ) 3 = 0.2398040 ⋅ π = 0.753366624 (40)

Regular Pentagonal Dodecahedron [5^{12}]:

V p = 5 2 φ 4 a 3 = 5 2 φ 3 ( 1 + φ 2 ) a 3 = 7.6631188998 ⋯ a 3 (41)

r i = 1 2 5 φ 5 a = 1 2 φ 2 ⋅ 1 + φ 2 a = 1.113516364 ⋯ a (42)

V s p h = 4 3 π r i 3 = π ⋅ 1.840893008 ⋯ a 3 (43)

V s p h V p = 0.240227643 ⋅ π = 0.754697398 (44)

A p = 3 ⋅ 5 ( 5 + 2 5 ) ⋅ a 2 = 15 φ 1 + φ 2 a 2 = 20.64572881 ⋯ a 2 (45)

A s p h = 4 π r i 2 = π 1 5 φ 5 a 2 = π ⋅ 4.9596747 ⋯ a 2 (46)

A s p h A p = 0.240227643 ⋅ π = 0.754697398 (47)

Rhombic Dodecahedron [4^{12}]:

V p = 16 3 3 a 3 (48)

r i = 2 3 a (49)

V s p h = 4 3 π ( 2 3 ) 3 a 3 = π ⋅ 8 6 27 a 3 (50)

V s p h V p = π ⋅ 2 6 = π ⋅ 0.23570226 = 0.74048048 (51)

A p = 8 2 a 2 (52)

A s p h = 4 π 2 3 a 2 = π ⋅ 8 3 a 2 (53)

A s p h A p = π ⋅ 1 3 2 = π ⋅ 0.23570226 (54)

Octahedron [3^{8}]:

V p = 2 3 a 3 (55)

r i = a 6 (56)

V s p h = 4 3 π ( a 6 ) 3 (57)

V s p h V p = π 1 3 3 = π ⋅ 0.192450089 = 0.604599788 (58)

A p = 2 3 a 2 (59)

A s p h = 4 π ( a 6 ) 2 = 2 3 π (60)

A s p h A p = π 1 3 3 (61)

Hexahedron (Cube) [4^{6}]:

V p = a 3 (62)

V s p h = π 6 a 3 (63)

V s p h V p = π 6 = 0.523598775 (64)

A p = 6 a 2 (65)

A s p h = 4 π ( a 2 ) 2 = π a 2 (66)

A s p A p = π 6 (67)

Tetrahedron [3^{4}] (self-dual polyhedron):

V p = 2 12 a 3 (68)

r i = 1 2 6 a (69)

V s p h = 4 3 π r i 3 = 4 3 π ( 1 2 6 ) 3 a 3 (70)

V s p h V p = π 3 ⋅ 6 = π ⋅ 0.096225044 = 0.30229989 (71)

A p = 3 a 2 (72)

A s p h = 4 π ( 1 2 6 ) 2 a 2 = π 6 a 2 (73)

A s p h A p = π 6 3 (74)

Equilateral Quadratic Pyramid (Half-Octahedron) [3^{4}4^{1}]:

V p = a 3 2 ⋅ 3 (75)

r i = a 2 ( 1 + 3 ) (76)

V s p h = 4 3 π r i 3 = 2 π a 3 3 ( 1 + 3 ) 3 (77)

V s p h V p = π ⋅ 2 ( 1 + 3 ) 3 = π ⋅ 0.0980762 = 0.308115505 (78)

A p = ( 1 + 3 ) a 2 (79)

A s p = 4 π a 2 2 ( 1 + 3 ) 2 (80)

A s p h A p = 2 π ( 1 + 3 ) 3 = π ⋅ 0.0980762 = 0.308115505 (81)

Great Pyramid [3^{4}4^{1}]:

In case of the Great Pyramid we use exceptionally for the basis length the value a = 2 in accordance with a recently published contribution [

V p = 1 3 a 2 h = 4 3 φ − 1 / 2 (82)

r i = φ 3 / 2 (83)

V s p h = 4 3 π r 3 = 4 3 π φ 9 / 2 (84)

V s p h V p = π ⋅ φ 5 = π ⋅ 0.0901699 = 0.283277 (85)

A p = 4 φ − 1 + 4 = 4 φ − 2 (86)

A s p h = 4 π r 2 = 4 π φ 3 (87)

A s p h A p = π ⋅ φ 5 (88)

In _{V} is depicted in relation to the number of faces. However, because one needs a minimum of 4 faces to put up a solid, the number of faces was reduced by 3. Therefore, a highly reliable least squares fit was performed using the relation

R V = 1 − exp ( − b ⋅ n F − 3 ) (89)

The parameter b was found to be 0.418 ± 0.011 (2.6%). This value approximates 2 − 1 = 0.414235 ⋯ known as silver mean. The tetragonal pyramids (half-octahedron and Great Pyramid) were not included in the fit, because they don’t match the regression curve (see green marked solids in

of the plot does not give a profound improvement.

Applying Equation (89) to the case of a hexakisicosahedron with 120 triangular faces, one can predict for its volume ratio a value of R V ≈ 0.99 . In practice, the polyhedral void space of polyhedral networks can approximately be determined by this formula.

Turning back to the deviation of the volume ratios for both quadratic pyramids from that of the fitted curve, denoted by R ′ V , one can determine the following approximations

Half-octahedron: R ′ V 2 R V = 1.0229 ≈ 1 (90)

Great Pyramid: 5 R ′ V 8 R V = 0.9834 ≈ 1 (91)

Interestingly, the term 5/8 in equation (90) approximates the golden mean φ.

However, symmetry reduction from icosahedral respectively cubic symmetry to tetragonal one resulted in the degeneration of the in-sphere to a biaxial ellipsoid as approximated void space. The volume V_{e} of a biaxial ellipsoid with z being the rotation axis is given by

V e = 4 3 π r x 2 r z (92)

where r_{x} and r_{z} are the corresponding half-axis.

Equilateral Quadratic Pyramid

In case of the half-octahedral pyramid the inscribed rotation ellipsoid of maximum volume has half-diameters of r z = r i − Δ max and r x = r i + Δ max ⋅ 2 .

Replacing these relations in Equation (92), then the first derivative d V d Δ = 0 delivers

Δ max = 2 − 2 / 2 3 r i = 0.430964 ⋅ r i . (93)

The volume ratio is represented in

V e / V p = V s p h V p ( 4 27 ( 5 + 7 2 ) ) = 0.45417 . (94)

This value fits relation (89) reasonable well. The numerical eccentricity of the maximum biaxial in-ellipsoid with k = r z r x yielded ϵ = 1 − k 2 = 0.93539 .

Great Pyramid

Turning to the comparable results for the Great Pyramid, one has to work with ellipsoid half-diameters r z = r i − Δ max and r x = r i + Δ max / φ . Now Δ max is calculated to be

Δ max = 2 − φ 3 r i = 0.4046162 ⋅ r i (95)

giving for the half diameters

r x = 2 3 φ ( 1 + φ ) = 0.735934838 , r z = 1 3 φ 3 2 ( 1 + φ ) = 0.28927809 (96)

and for its ratio k respectively eccentricity ε

k = r z r x = φ 2 = 0.393075688 , ε = 1 − k 2 = 4 − φ 2 = 0.919506 (97)

Then the volume V_{e} simply results in

V e = ( 2 3 ) 4 φ 7 2 ( 1 + φ ) 3 ⋅ π = 0.6562705 (98)

Using relation (82) the ratio V e V p (represented in

V e V p = 4 27 φ 4 ( 1 + φ ) 3 ⋅ π = 0.386946 (99)

This Great Pyramid volume ratio based on its maximum in-ellipsoid is still not quite well adapted to curve (89) in

Finally one should note that if the sphere degenerates to an ellipsoid, Equation

(4) no longer holds, because

A e A p > V s p h V p (100)

Again the case of the Great Pyramid should serve to demonstrate this. The surface A of a biaxial ellipsoid with rotation axis z and eccentricity ϵ = 1 − k 2 , k = r z r x is given by

A = 2 π r x 2 ( 1 + k 2 atanh ( ε ) ε ) (101)

The surface at maximum volume of the ellipsoid A_{e} is then calculated as A e = 2.093982 and

the ratio A e A p = 0.323538 > A s p h A p = π ⋅ φ 5 = 0.283277 (102)

Bilinski B20 Zonohedron

Finally, the in-ellipsoid approach was applied to the Bilinski B20 zonohedron as example for an oblate polyhedron with a unique five-fold axis, derived from the triacontahedron by removing a complete zone of 10 faces. The calculation of its in-ellipsoid half-diameters was given in the Appendix. Then one determines the ellipsoid volume according to Equation (92) as

V e l = 5.165 a 3 (103)

The volume of the B20 polyhedron is one half of that of the triacontahedron

V p = 2 5 + 2 5 a 3 = 6.155367 a 3 (104)

and the volume ratio, depicted by the top blue cross in

R ′ V = V e l V p = 0.8391 (105)

Contrary, the calculation of the volume ratio using an in-sphere radius defined by Equation (1) leads to a very low ratio, with A p = 2 3 A p ( triacontahedron )

r i = 3 V p A p = 3 4 r i ( triacontahedron ) a = 1.032286 a (106)

V s p h = 4 3 π r i 3 a 3 = 9 π 16 ( 1 + 2 5 ) 3 / 2 = 4.6077539 a 3 (107)

R V = V s p h V p = 0.748575 (108)

Information about the in-sphere of polyhedral solids as interstitial void space in networks of alumo-silicates (zeolites), metallic glasses, amorphous alloys or liquids are important to understand their packing density and related properties such as phase transitions [

The relationship between volume and surface goes far beyond a purely geometric understanding. The duality between volume and surrounding surface respectively between any compact entity and assigned surface in general as well as the duality between a moving particle respectively body and the accompanying wave or reciprocity between matter and dark matter is the very spice of life. It has been impressively formulated by the words of Nobel laureate Wolfgang Pauli: “God made the bulk; surfaces were invented by the devil” (quoted from [

This contribution was inspired by a flaw in a recently published own paper. It compares the ratio of in-sphere volume to polyhedral volume of selected convex solids, which is always equal to the corresponding ratio of in-sphere surface and polyhedral surface. The volume ratio of the Great Pyramid is the smallest among the selected solids and also smaller as the similar half-octahedral pyramid as well as the tetrahedron. Convex solids with icosahedral symmetry indicate a slightly higher volume ratio than solids with cubic symmetry respectively tetragonal symmetry. However, replacing the in-sphere by the maximum inscribed biaxial ellipsoid one can adjust the volume ratio more reliably to a proposed relationship that connects the volume ratio with the number of faces. The in-sphere volume ratio of the Great Pyramid was recently shown to be π⋅φ^{5}, where φ is the golden mean. Assigned as fundamental number of nature, the fifth power of the golden mean, on the other hand, is fascinating through its connection to phase transition from microscopic to cosmic scale and may become important in connection with the DNA genetic code.

This number is also indirectly found in the expressions for the in-sphere surface of the regular pentagonal dodecahedron, not really expected in the term for the volume ratio of the hexakishexahedron, and as an approximation in the in-sphere volume relation for the soccer ball respectively the triacontahedron. Besides a solely geometrical interest, the theory of random packing respectively the determination of polyhedral void space as in-sphere volume respectively in-ellipsoid volume is important for practical applications.

I am indebted to P. P. Klein from the TU Clausthal, who made me known to a flaw in the Appendix of my recent work [

The author declares no conflicts of interest regarding the publication of this paper.

Otto, H.H. (2021) Ratio of In-Sphere Volume to Polyhedron Volume of the Great Pyramid Compared to Selected Convex Polyhedral Solids. Journal of Applied Mathematics and Physics, 9, 41-56. https://doi.org/10.4236/jamp.2021.91005

Truncated Octahedron:

The distances r i , f from the center to the different faces f(square and hexagon) are calculated to be r i , 4 = 2 ⋅ a and r i , 6 = 1 2 6 ⋅ a . Now we use a weighted

in-sphere radius, where the face areas are thought to be a measure of the radius of curvature. Then we get the weighted radius as

r i = 2 + 2 3 ⋅ 1 2 6 1 + 2 3 a = 4 2 1 + 2 3 a = 1.267187608 ⋅ a (109)

This weighted radius is identical to that used in Equation (35) derived by application of the relation r i = 3 V p A p underlining the universal importance of this

relation. The volume ratio of the truncated octahedron is well represented by Equation (89) using this approach.

Triacontahedron (FigureA1):

It belongs to the icosahedral group I_{h} and is a regular polyhedron consisting of 30 equilateral rhobus faces (golden rombi), 32 vertices and 60 edges. It was first discovered by Johannes Kepler (1571-1630). Many years ago I had chosen this solid serving as a structural model forthe first building sphere of quasi-crystals belonging to the Al-Mn alloy system, where neighboring polyhedra interpenetrate [

Interestingly, the volume ratio of the triacontahedron can by approximated by [

V s p h V p = 4 3 π r i 3 = π ⋅ φ − 3 15 = 0.8872000 ≈ 2 25 ⋅ φ 5 = 0.88721359 (110)

In this way, one is again faced with the fifth power of the golden mean.

Bilinski Zonohedron

If one remove a complete zone from the rhombic triacontahedron, for instance the middle zone parallel to the c axis in FigureA1, then an oblate solid with 20 faces results named Bilinski’s B20 zonohedron [

has figured out by Cowley in 1752 [

Using two independent Cartesian face coordinates of the B20 zonohedron, choosing faces of different steepness, on can solve the ellipsoid equation to get the two half diameters r_{x} and r_{z} of the biaxial ellipsoid as

r x = z 2 2 ( x 1 2 + y 1 2 ) − z 1 2 ( x 2 2 + y 2 2 ) z 2 2 − z 1 2 (111)

r z = r x z 1 r x 2 − x 1 2 − y 1 2 (112)

Coordinates of the triacontahedron given by Gray [

x 1 = ϕ 5 a , y 1 = 0 , z 1 = 3 2 5 a ; (113)

x 2 = 1 5 ( ϕ + 1 2 ) a , y 2 = 1 2 1 + 2 5 a , z 2 = 1 2 5 a (114)

With these coordinates one yields the B20 in-ellipsoid half-diameters

r x = 1.21520448 a and r z = 0.834993 a . (115)