Mathematical analysis of a small rhombicuboctahedron (Archimedean solid) by HCR

Mathematical Analysis of Small Rhombicuboctahedron/Archimedean solid
Application of HCR’s formula for regular polyhedrons (all five platonic solids)
Applications of “HCR’s Theory of Polygon” proposed by Mr H.C. Rajpoot (year-2014)
©All rights reserved
Mr Harish Chandra Rajpoot
M.M.M. University of Technology, Gorakhpur-273010 (UP), India Dec, 2014
Introduction: A small rhombicuboctahedron is an Archimedean solid which has 8 congruent equilateral
triangular & 18 congruent square faces each having equal edge length. It is created/generated either by
shifting/translating all 8 equilateral triangular faces of a regular octahedron radially outwards by the same
distance without any other transformation (i.e. rotation, distortion etc.) or by shifting/translating all 6 square
faces of a cube (regular hexahedron) radially outwards by the same distance without any other transformation
(i.e. rotation, distortion etc.) till either the vertices, initially coincident, of each four equilateral triangular faces
of the octahedron form a square of the same edge length or the vertices, initially coincident, of each three
square faces of the cube form an equilateral triangle of the same edge length. Both the methods create the
same solid having 8 congruent equilateral triangle & 18 congruent square faces each having equal edge
length. This solid is called small rhombicuboctahedron which is an Archimedean solid. For calculating all the
parameters of a small rhombicuboctahedron, we would use the equations of right pyramid & regular
octahedron.
Radial expansion of a regular octahedron: For ease of calculations, let there be a regular octahedron with
edge length & its centre at the point O. Now all its 8 equilateral triangular faces are shifted/translated
radially outward by the same distance without any other transformation (i.e. rotation, distortion etc.) till the
vertices, initially coincident, of each four triangular faces of the octahedron form a square of the same edge
length to obtain a small rhombicuboctahedron along with 18 additional square faces of the same edge
length . (See figure 1 which shows an equilateral triangular face & a square face with a common vertex A &
their normal distances respectively from the centre O of the parent octahedron).
Angle ( ) between the consecutive lateral edges of any of the elementary right pyramids of
parent octahedron: We know that the angle ( ) between any two consecutive lateral edges of any of the
elementary right pyramids of any regular polyhedron (all five platonic solids) is given by HCR’s formula for
platonic solids (to calculate edge angle ) as follows
( √ { } {
( )
})
In this case of a regular octahedron, we have
Now, setting both these integer values in HCR’s Formula, we get

( √ { } {
( )
}) ( √ { } { })
( √ ) (√ √ )
(√ √ ( ) ( )) (√ √ )
(√ ( ( )) ) (√ ) ( )
⇒ ⇒
( )
Angle ( ) between the normal axis & the lateral edge of any of the elementary right pyramids
with equilateral triangular base (i.e. face of parent octahedron): We know that the angle ( ) between
the normal axis & the lateral edge of any right pyramid with regular n-polygonal base & an angle between
consecutive lateral edges is given by the following formula (taken from the eq(V) in “Two Mathematical
proofs of Bond Angle in Regular Tetrahedral Structure” by HCR)
√
Now substituting the value of ⁄ from eq(I) in the above expression, we get
√( ) ( )
√
√
√
√
√
√
(√ ) ( )
The above result showing the angle between the normal axis OM of equilateral triangular face & the normal
axis ON of square face which remains constant for translation of all the faces of parent octahedron without
any other transformation (i.e. rotation, distortion etc. ) (see figure 1 below)
Derivation of the outer (circumscribed) radius ( ) of small rhombicuboctahedron:
Let be the radius of the spherical surface passing through all 24 vertices of a given small
rhombicosidodecahedron with 8 congruent equilateral triangular & 18 congruent square faces each of edge
length . Now consider any of the equilateral triangular faces & any of three adjacent square faces each of
edge length & common vertex A. (see figure 1 showing a sectional view of the adjacent triangular & square
faces with a common vertex A)

⇒
√ √
⇒
√
√
In right
⇒
(
√
)
√
(
√
)
In right
⇒
(
√
)
√
(
√
)
Since angle is constant for the pure translation of the equilateral triangular faces of the parent octahedron
hence we have the following condition
Now, substituting the corresponding values of in the above expression, we have
(
√
) (
√
) (√ )
⇒ (
√
√ (
√
)
√
√ (
√
) )
(
√
√ (√ ) )
( ( √ √ ) (
√
))
⇒
√
√
√
√ √
Figure 1: An equilateral triangular face with centre M & a
square face with centre N having a common vertex A & the
normal distances 𝑯 𝑻 𝑯 𝒔 respectively from the centre O of
small rhombicuboctahedron. Angle 𝜷 between the normal
axis OM of equilateral triangular face & the normal axis ON of
square face remains constant for the translation of the faces
of a regular octahedron.

⇒ (
√
√
√
√ ) (√ ) ( )
⇒ ( ) ( )
√
√
√( ) ( )
⇒ ( ) ( )
√
√
√( ) ( )
( ) ( )
√
√
√( ) ( )
√
√
√
⇒ ( ) (
√
√
√ )
⇒ ( )
⇒
⇒
⇒
Now, solving the above quadratic equation for the values of K as follows
⇒
( ) √( )
√ √ √ √
1. Taking positive sign, we have
√
( )
Hence, above value is not acceptable.
2. Taking negative sign, we have

√
Hence, above value is accepted, now we have
√
( √ )
√
( √ )
√
( √ )
( √ )( √ )
√
( √ )
√ √
Hence, the outer (circumscribed) radius ( ) of a small rhombicuboctahedron with edge length is given as
√ √ ( )
Normal distance ( ) of equilateral triangular faces from the centre of small
rhombicuboctahedron: The normal distance ( ) of each of 8 congruent equilateral triangular faces from
the centre O of a small rhombicuboctahedron is given as
√( ) ( ) ( )
⇒ √( √ √ ) (
√
) √ √ √ √ √( √ ) ( √ )
√
⇒
( √ )
√
( )
It’s clear that all 8 congruent equilateral triangular faces are at an equal normal distance from the centre
of any small rhombicuboctahedron.
Solid angle ( ) subtended by each of the equilateral triangular faces at the centre of small
rhombicuboctahedron: we know that the solid angle ( ) subtended by any regular polygon with each side
of length at any point lying at a distance H on the vertical axis passing through the centre of plane is given by
“HCR’s Theory of Polygon” as follows
(
√
)
Hence, by substituting the corresponding values in the above expression, we get the solid angle subtended by
each equilateral triangular face at the centre of small rhombicuboctahedron as follows
(
(
( √ )
√
)
√ (
( √ )
√
)
)

(
( √ )
√
√
√ √
)
(
√ √
√ √
) ( √
√
( √ )
)
( √
( √ )( √ )
( √ )( √ )
) ( √ √
) ( √ √ )
( √ √ ) ( )
Normal distance ( ) of square faces from the centre of small rhombicuboctahedron: Similarly, the
normal distance ( ) of each of 18 congruent square faces from the centre of small rhombicuboctahedron is
given as
√( ) ( ) ( )
⇒ √( √ √ ) (
√
) √ √ √ √ √( √ ) ( √ )
⇒
( √ )
( )
It’s clear that all 18 congruent square faces are at an equal normal distance from the centre of any small
rhombicuboctahedron.
Solid angle ( ) subtended by each of the square faces at the centre of small
rhombicuboctahedron: we know that the solid angle ( ) subtended by a square with each side of length
at any point lying at a distance H on the vertical axis passing through the centre of plane is given by “HCR’s
Theory of Polygon” as follows
( )
Hence, by substituting the corresponding values in the above expression, we get the solid angle subtended by
each square face at the centre of small rhombicuboctahedron as follows
(
(
( √ )
)
)
(
√
) (
( √ )
) (
( √ )
)
(
√
) ( )
It’s clear from the above results that the solid angle subtended by each of 18 square faces is greater than the
solid angle subtended by each of 8 equilateral triangular faces at the centre of any small

It’s also clear from eq(II) & (IV) that i.e. the normal distance ( ) of equilateral triangular faces is
greater than the normal distance of the square faces from the centre of a small rhombicuboctahedron i.e.
square faces are the closer to the centre as compared to the equilateral triangular faces in any small
Important parameters of a small rhombicuboctahedron:
1. Inner (inscribed) radius ( ): It is the radius of the largest sphere inscribed (trapped inside) by a
small rhombicuboctahedron. The largest inscribed sphere always touches all 18 congruent square
faces but does not touch any of 8 congruent equilateral triangle faces at all since all 18 square faces
are closer to the centre as compared to all 8 triangular faces. Thus, inner radius is always equal to the
normal distance ( ) of square faces from the centre of a small rhombicuboctahedron & is given from
the eq(IV) as follows
( √ )
Hence, the volume of inscribed sphere is given as
( ) (
( √ )
)
2. Outer (circumscribed) radius ( ): It is the radius of the smallest sphere circumscribing a small
rhombicuboctahedron or it’s the radius of a spherical surface passing through all 24 vertices of a small
rhombicuboctahedron. It is from the eq(I) as follows
√ √
Hence, the volume of circumscribed sphere is given as
( ) ( √ √ )
3. Surface area ( ): We know that a small rhombicuboctahedron has 8 congruent equilateral
triangular & 18 congruent square faces each of edge length . Hence, its surface area is given as
follows
( ) ( )
We know that area of any regular n-polygon with each side of length is given as
Hence, by substituting all the corresponding values in the above expression, we get
( ) ( )
√
( √ )
( √ )
4. Volume ( ): We know that a small rhombicuboctahedron with edge length has 8 congruent
equilateral triangular & 18 congruent square faces. Hence, the volume (V) of the small

rhombicuboctahedron is the sum of volumes of all its elementary right pyramids with equilateral
triangular & square bases (faces) given as follows
( )
( )
( ( ) ) ( ( ) )
( ( )
( √ )
√
) ( ( )
( √ )
)
( √ )
( √ )
( √ √ ) ( √ ) ( √ )
( √ )
5. Mean radius ( ): It is the radius of the sphere having a volume equal to that of a small
rhombicuboctahedron. It is calculated as follows
( )
( √ )
⇒ ( )
( √ )
(
√
)
(
√
)
It’s clear from above results that
Construction of a solid small rhombicuboctahedron: In order to construct a solid small
rhombicuboctahedron with edge length there are two methods
1. Construction from elementary right pyramids: In this method, first we construct all elementary right
pyramids as follows
Construct 8 congruent right pyramids with equilateral triangular base of side length & normal height ( )
( √ )
√
Construct 18 congruent right pyramids with square base of side length & normal height ( )
( √ )
Now, paste/bond by joining all these elementary right pyramids by overlapping their lateral surfaces & keeping
their apex points coincident with each other such that all the edges of each equilateral triangular base (face)
coincide with the edges of three square bases (faces). Thus a solid small rhombicuboctahedron, with 8
congruent equilateral triangular & 18 congruent square faces each of edge length , is obtained.

2. Facing a solid sphere: It is a method of facing, first we select a blank as a solid sphere of certain material
(i.e. metal, alloy, composite material etc.) & with suitable diameter in order to obtain the maximum desired
edge length of a small rhombicuboctahedron. Then, we perform the facing operations on the solid sphere to
generate 8 congruent equilateral triangular & 18 congruent square faces each of equal edge length.
Let there be a blank as a solid sphere with a diameter D. Then the edge length , of a small
rhombicuboctahedron of the maximum volume to be produced, can be co-related with the diameter D by
relation of outer radius ( ) with edge length ( ) of the small rhombicuboctahedron as follows
√ √
Now, substituting ⁄ in the above expression, we have
√ √
√ √
√ √
Above relation is very useful for determining the edge length of a small rhombicuboctahedron to be
produced from a solid sphere with known diameter D for manufacturing purpose.
Hence, the maximum volume of small rhombicuboctahedron produced from a solid sphere is given as follows
( √ ) ( √ )
(
√ √
)
( √ )
( √ )√ √
( √ )( √ )
√ √
( √ )
√ √
( √ )
√ √
Minimum volume of material removed is given as
( ) ( )
( )
( √ )
√ √
(
( √ )
√ √
)
( ) (
( √ )
√ √
)
Percentage ( ) of minimum volume of material removed

(
( √ )
√ √
)
(
( √ )
√ √
)
It’s obvious that when a small rhombicuboctahedron of the maximum volume is produced from a solid
sphere then about of material is removed as scraps. Thus, we can select optimum diameter of blank
as a solid sphere to produce a solid small rhombicuboctahedron of the maximum volume (or with maximum
desired edge length)
Conclusions: Let there be any small rhombicuboctahedron having 8 congruent equilateral triangular & 18
congruent square faces each with edge length then all its important parameters are
calculated/determined as tabulated below
Congruent
polygonal faces
No. of
faces
Normal distance of each face from the centre
of the small rhombicuboctahedron
Solid angle subtended by each face at the centre
of the small rhombicuboctahedron
Equilateral
triangle
8 ( √ )
√
( √ √ )
Square 18 ( √ )
(
√
)
Inner (inscribed) radius ( ) ( √ )
Outer (circumscribed) radius ( )
√ √
Mean radius ( ) (
√
)
Surface area ( ) ( √ )
Volume ( ) ( √ )
Note: Above articles had been developed & illustrated by Mr H.C. Rajpoot (B Tech, Mechanical Engineering)
M.M.M. University of Technology, Gorakhpur-273010 (UP) India Dec, 2014
Email: rajpootharishchandra@gmail.com
Author’s Home Page: https://notionpress.com/author/HarishChandraRajpoot

Mathematical analysis of a small rhombicuboctahedron (Archimedean solid) by HCR

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