Mathematical analysis of small rhombicosidodecahedron (Archimedean solid) by hcr
13
Mathematical Analysis of Small Rhombicosidodecahedron/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 rhombicosidodecahedron is an Archimedean solid which has 20 congruent equilateral triangular, 3o congruent square & 12 congruent regular pentagonal faces each having equal edge length. It is created/generated either by shifting/translating all 20 equilateral triangular faces of a regular icosahedron radially outwards by the same distance without any other transformation (i.e. rotation, distortion etc.) or by shifting/translating all 12 pentagonal faces of a regular dodecahedron radially outwards by the same distance without any other transformation (i.e. rotation, distortion etc.) till either the vertices, initially coincident, of each five triangular faces of the icosahedron form a regular pentagon of the same edge length or the vertices, initially coincident, of each three regular pentagonal faces of the dodecahedron form an equilateral triangle of the same edge length. Both the methods create the same solid having 20 congruent equilateral triangle, 3o congruent square & 12 congruent regular pentagonal faces each having equal edge length. This solid is called small rhombicosidodecahedron which is an Archimedean solid. For calculating all the parameters of a small rhombicosidodecahedron, we would use the equations of right pyramid & regular icosahedron. Radial expansion of a regular icosahedron: For ease of calculations, let there be a regular icosahedron with edge length & its centre at the point O. Now all its 20 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 five triangular faces of icosahedron form a regular pentagon of the same edge length to obtain a small rhombicosidodecahedron along with 30 additional square faces of the same edge length . (See figure 1 which shows an equilateral triangular face & a regular pentagonal face with a common vertex A & their normal distances respectively from the centre O of the parent icosahedron). Angle () between the consecutive lateral edges of any of the elementary right pyramids of parent icosahedron: 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 icosahedron, we have Now, substituting both these integer values in HCR’s Formula, we get
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Transcript of Mathematical analysis of small rhombicosidodecahedron (Archimedean solid) by hcr
Mathematical Analysis of Small Rhombicosidodecahedron/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 rhombicosidodecahedron is an Archimedean solid which has 20 congruent equilateral
triangular, 3o congruent square & 12 congruent regular pentagonal faces each having equal edge length. It is
created/generated either by shifting/translating all 20 equilateral triangular faces of a regular icosahedron
radially outwards by the same distance without any other transformation (i.e. rotation, distortion etc.) or by
shifting/translating all 12 pentagonal faces of a regular dodecahedron radially outwards by the same distance
without any other transformation (i.e. rotation, distortion etc.) till either the vertices, initially coincident, of
each five triangular faces of the icosahedron form a regular pentagon of the same edge length or the vertices,
initially coincident, of each three regular pentagonal faces of the dodecahedron form an equilateral triangle of
the same edge length. Both the methods create the same solid having 20 congruent equilateral triangle, 3o
congruent square & 12 congruent regular pentagonal faces each having equal edge length. This solid is called
small rhombicosidodecahedron which is an Archimedean solid. For calculating all the parameters of a small
rhombicosidodecahedron, we would use the equations of right pyramid & regular icosahedron.
Radial expansion of a regular icosahedron: For ease of calculations, let there be a regular icosahedron
with edge length & its centre at the point O. Now all its 20 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 five triangular faces of icosahedron form a regular pentagon of the same
edge length to obtain a small rhombicosidodecahedron along with 30 additional square faces of the same
edge length . (See figure 1 which shows an equilateral triangular face & a regular pentagonal face with a
common vertex A & their normal distances respectively from the centre O of the parent
icosahedron).
Angle ( ) between the consecutive lateral edges of any of the elementary right pyramids of
parent icosahedron: 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 icosahedron, we have
Now, substituting both these integer values in HCR’s Formula, we get
Mathematical Analysis of Small Rhombicosidodecahedron/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
(
√ {
} {
( )
}) ( √ {
} {
})
( √ ) (√ √ )
(√ √ ( ) ( )) (√ √ ( ) )
(√ √ ) (√ √
√
) (√ √
√
)
(√ √
) (√(√ )
, (
√
)
(√
) (
√
) ( )
Angle ( ) between the normal axis & the lateral edge of any of the elementary right pyramids
with equilateral triangular base (i.e. face of parent icosahedron): 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 regular pentagonal face which remains constant for translation of all the faces of parent
icosahedron without any other transformation (i.e. rotation, distortion etc. ) (see figure 1 below)
Derivation of the outer (circumscribed) radius ( ) of small rhombicosidodecahedron:
Let be the radius of the spherical surface passing through all the vertices of a given small
rhombicosidodecahedron with 20 congruent equilateral triangular, 3o congruent square & 12 congruent
regular pentagonal faces each of edge length . Now consider any of the equilateral triangular faces & any of
Mathematical Analysis of Small Rhombicosidodecahedron/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
three adjacent regular pentagonal faces each of edge length & common vertex A. (see figure 1 showing a
sectional view of the adjacent triangular & pentagonal faces with a common vertex A)
⇒
√
√
⇒
In right
⇒
(
√ *
√
(
√
)
In right
⇒
(
)
(
*
Since angle is constant for the pure translation of the equilateral triangular faces of parent icosahedron
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 regular pentagonal face with centre N having a common
vertex A & the normal distances 𝑯𝑻 𝑯𝑷 respectively from the centre O of small rhombicosidodecahedron. Angle 𝜷 between the normal axis OM of equilateral triangular face & the normal axis ON of regular pentagonal face remains constant for the translation of the faces of a regular icosahedron.
Mathematical Analysis of Small Rhombicosidodecahedron/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
⇒
√
√
√
( √ )
√ √
⇒ (
√
√
√
)
(( √ )
√ √ )
( )
⇒
(
)
(
)
√
√(
)(
)
√
√
√
√
√
⇒
(
*
(
*
√ √(
* (
*
⇒ (
(
*
(
* *
(
√ √(
* (
*)
⇒
(
)
(
)
(
)
(
*
(
*
(
)
⇒ (
* (
*
On setting the values of
(
(√ √
+
( √
)
(√ √
+
(√ √
+
)
( √
)
(
(√ √
+
)
( √
)
⇒ (
( √ )
√
) (
√
)(
√
)
√
⇒ (
√
√
) (
√
)(
√
)
√
Mathematical Analysis of Small Rhombicosidodecahedron/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
⇒ ( √
) (
√
)
( √ )
Now, solving above quadratic equation for the value 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, outer (circumscribed) radius ( ) of a small rhombicosidodecahedron with edge length is given as
√ √ ( )
Normal distance ( ) of equilateral triangular faces from the centre of small
rhombicosidodecahedron: The normal distance ( ) of each of 20 congruent equilateral triangular faces
from the centre O of small rhombicosidodecahedron is given as
√( ) ( ) ( )
Mathematical Analysis of Small Rhombicosidodecahedron/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
⇒ √(
√ √ )
(
√ *
√ √
√
√
√( √ )
( √ )
√
⇒ ( √ )
√ ( )
It’s clear that all 20 congruent equilateral triangular faces are at an equal normal distance from the
centre of any small rhombicosidodecahedron.
Solid angle ( ) subtended by each of the equilateral triangular faces at the centre small
rhombicosidodecahedron: 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 rhombicosidodecahedron as follows
(
(
( √ )
√ )
√ (( √ )
√ )
)
(
( √ )
√
√
√ √
)
(√ √
√ √ + (
√
√
√ )
(
√
( √ )( √ )
( √ )( √ )) (
√
√
)
(
√
√
) ( )
Normal distance ( ) of regular pentagonal faces from the centre of small
rhombicosidodecahedron: The normal distance ( ) of each of 12 congruent regular pentagonal faces
from the centre O of small rhombicosidodecahedron is given as
√( ) ( ) ( )
Mathematical Analysis of Small Rhombicosidodecahedron/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
⇒ √(
√ √ )
(
)
√ √
( √ )
√ √
( √ )
√
√ √
√
√
√
√
⇒
√
√
( )
It’s clear that all 12 congruent regular pentagonal faces are at an equal normal distance from the centre
of any small rhombicosidodecahedron.
Solid angle ( ) subtended by each of the regular pentagonal faces at the centre of small
rhombicosidodecahedron: we know that the solid angle ( ) subtended by any regular polygon is given by
“HCR’s Theory of Polygon” as follows
(
√ )
Hence, by substituting the corresponding value of normal distance in the above expression, we get the
solid angle subtended by each regular pentagonal face at the centre of small rhombicosidodecahedron as
follows
(
(
√ √
)
√ (
√ √
)
)
( )
(
√ √
√ √
√ √
√
√ )
(√ √ √
√ √ √ +
(√( √ )( √ )
( √ )( √ )) (√
√
)
(√ √
) (
√
√
)
(
√
√
) ( )
Mathematical Analysis of Small Rhombicosidodecahedron/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
Normal distance ( ) of square faces from the centre of small rhombicosidodecahedron: Similarly,
the normal distance ( ) of each of 30 congruent square faces from the centre of small
rhombicosidodecahedron is given as
√ ( )
⇒ √(
√ √ )
(
√ *
√ √
√
√
√( √ )
( √ )
⇒ ( √ )
( )
It’s clear that all 30 congruent square faces are at an equal normal distance from the centre of any small
rhombicosidodecahedron.
Solid angle ( ) subtended by each of the square faces at the centre of small
rhombicosidodecahedron: 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 rhombicosidodecahedron as follows
(
(( √ )
)
)
(
√ * (
√
) (
√
)
( √
) ( )
It’s clear from the above results that the solid angle subtended by each of the regular pentagonal faces is
greater than the solid angle subtended by each of the equilateral triangular faces & each of the square faces at
the centre of any small rhombicosidodecahedron.
It’s also clear from eq(II), (IV) & (VI) i.e. the normal distance ( ) of equilateral triangular faces
is greater than the normal distances of square faces & regular pentagonal faces from the centre of
the small rhombicosidodecahedron i.e. pentagonal faces are the closer to the centre as compared to the
equilateral triangular faces & the square faces in any small rhombicosidodecahedron.
Important parameters of a small rhombicosidodecahedron:
1. Inner (inscribed) radius( ): It is the radius of the largest sphere inscribed (trapped inside) by the
small rhombicosidodecahedron. The largest inscribed sphere always touches all 12 congruent regular
pentagonal faces but does not touch any of 20 congruent equilateral triangle & 30 congruent square
faces at all since all 12 pentagonal faces are closer to the centre as compared to all 20 triangle & 30
square faces. Thus, inner radius is always equal to the normal distance ( ) of regular pentagonal
faces from the centre of a small rhombicosidodecahedron & is given from the eq(IV) as follows
Mathematical Analysis of Small Rhombicosidodecahedron/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
√
√
Hence, the volume of inscribed sphere is given as
( )
(
√
√
)
2. Outer (circumscribed) radius( ): It is the radius of the smallest sphere circumscribing a given
small rhombicosidodecahedron or it’s the radius of a spherical surface passing through all 60 vertices
of a given small rhombicosidodecahedron. 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 rhombicosidodecahedron has 20 congruent equilateral
triangular, 30 congruent square & 12 congruent regular pentagonal 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 rhombicosidodecahedron with edge length has 20 congruent
equilateral triangular, 3o congruent square & 12 congruent regular pentagonal faces. Hence, the
volume (V) of the small rhombicosidodecahedron is the sum of volumes of all its elementary right
pyramids with equilateral triangular, square & regular pentagonal bases (faces) given as follows
( )
( )
( )
(
( ) * (
( ) *
(
( ) *
Mathematical Analysis of Small Rhombicosidodecahedron/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
(
(
*
( √ )
√ ) (
(
*
( √ )
)
(
(
*
√
√
)
( √ )
( √ )
√
( √ )( √ )
( √ )
( ( √ )
( √ )
√
( √ )
( √ ))
( ( √ )
√
( √ )( √ )
( √ )( √ ))
( ( √ )
√
√
) (
( √ )
√
√
)
( ( √ )
√( √ )
, (
( √ )
√ ( √ )
)
( √ √
) (
√
)
( √ )
( √ )
5. Mean radius( ): It is the radius of the sphere having a volume equal to that of a given small
rhombicosidodecahedron. It is calculated as follows
( )
( √ )
⇒ ( )
( √ )
(
√
)
( √
)
It’s clear from above results that
Construction of a solid small rhombicosidodecahedron: In order to construct a solid small
rhombicosidodecahedron 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
Mathematical Analysis of Small Rhombicosidodecahedron/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
Construct 20 congruent right pyramids with equilateral triangular base of side length & normal height ( )
( √ )
√
Construct 30 congruent right pyramids with square base of side length & normal height ( )
( √ )
Construct 12 congruent right pyramids with regular pentagonal 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) & all the edges of each regular pentagonal base (face)
coincide with the edges of five square bases (faces). Thus, a solid small rhombicosidodecahedron, with 20
congruent equilateral triangular, 30 congruent square & 12 congruent regular pentagonal 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 small rhombicosidodecahedron. Then, we perform facing operations on the solid sphere to
generate 20 congruent equilateral triangular, 30 congruent square & 12 congruent regular pentagonal 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
rhombicosidodecahedron 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 rhombicosidodecahedron as follows
√ √
Now, substituting ⁄ in the above expression, we have
√ √
√ √
√ √
Above relation is very useful for determining the edge length of a small rhombicosidodecahedron to be
produced from a solid sphere with known diameter D for manufacturing purpose.
Hence, the maximum volume of small rhombicosidodecahedron produced from the solid sphere is given as
follows
( √ )
( √ )
(
√ √ )
( √ )
( √ )√ √
Mathematical Analysis of Small Rhombicosidodecahedron/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
( √ )( √ )
√ √
( √ )
√ √
( √ )
√ √
Minimum volume of material removed is given as
( ) ( )
( )
( √ )
√ √ (
√
√ √ )
( ) (
√
√ √ )
Percentage ( ) of minimum volume of material removed
(
√
√ √ )
(
( √ )
√ √ )
It’s obvious that when a small rhombicosidodecahedron 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 rhombicosidodecahedron of the maximum volume (or with
maximum desired edge length)
Conclusions: let there be any small rhombicosidodecahedron having 20 congruent equilateral triangular
faces, 30 congruent square faces & 12 congruent regular pentagonal 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 given small rhombicosidodecahedron
Solid angle subtended by each face at the centre of the given small rhombicosidodecahedron
Equilateral triangle
20
( √ )
√
(
√
√
)
Square
30
( √ )
( √
)
Regular pentagon
12
√
√
(
√
√
)
Mathematical Analysis of Small Rhombicosidodecahedron/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
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: [email protected]
Author’s Home Page: https://notionpress.com/author/HarishChandraRajpoot
Courtesy: Advanced Geometry by Harish Chandra Rajpoot
