Hcr's Theory of Polygon (Proposed by Mr Harish Chandra Rajpoot)
Optimum Solution of Snub Cube (an Archimedean solid) by using "HCR's Theory of Polygon" &...
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Optimum solution for snub cube (Archimedean solid) Application of “HCR’s Theory of Polygon” & “Newton-Raphson Method” 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: Snub cube is a uniform polyhedron which is one of 13 Archimedean solids. A snub cube is created/generated first by radially expanding a cube to obtain the small rhombicuboctahedron & then partially rotating all 6 original square faces in such a way that each of 12 additional square faces is changed into two equilateral triangular faces each with edge length equal to that of the original cube & all 24 identical vertices exactly lie on a spherical surface. Analysis of snub cube: For ease of calculations, let there be a cube (regular hexahedron) with edge length & its centre at the point O. Now all its 6 square faces are first shifted/translated radially outward by the same distance & then partially rotated to obtain a snub cube with edge length & outer (circumscribed) radius . (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 snub cube) Derivation of the outer (circumscribed) radius of snub cube: Let be the radius of the spherical surface passing through all 24 vertices of a given snub cube with 32 congruent equilateral triangular & 6 congruent square faces each of edge length . Now consider any of the equilateral triangular faces & its adjacent square face of edge length & common vertex A. (see figure 1 showing a sectional view of the adjacent triangular & square faces with a common vertex A of the snub cube) In right ⇒ √ √ √ √ ( √ ) √ √ In right ⇒ √ √
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Transcript of Optimum Solution of Snub Cube (an Archimedean solid) by using "HCR's Theory of Polygon" &...
Optimum solution for snub cube (Archimedean solid) Application of “HCR’s Theory of Polygon” & “Newton-Raphson Method”
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: Snub cube is a uniform polyhedron which is one of 13 Archimedean solids. A snub cube is
created/generated first by radially expanding a cube to obtain the small rhombicuboctahedron & then partially
rotating all 6 original square faces in such a way that each of 12 additional square faces is changed into two
equilateral triangular faces each with edge length equal to that of the original cube & all 24 identical vertices
exactly lie on a spherical surface.
Analysis of snub cube: For ease of calculations, let there be a cube (regular hexahedron) with edge length
& its centre at the point O. Now all its 6 square faces are first shifted/translated radially outward by the same
distance & then partially rotated to obtain a snub cube with edge length & outer (circumscribed) radius .
(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 snub cube)
Derivation of the outer (circumscribed) radius of snub cube:
Let be the radius of the spherical surface passing through all 24 vertices of a given snub cube with 32
congruent equilateral triangular & 6 congruent square faces each of edge length . Now consider any of the
equilateral triangular faces & its adjacent square face of edge length & common vertex A. (see figure 1
showing a sectional view of the adjacent triangular & square faces with a common vertex A of the snub cube)
In right
⇒
√
√
√ √ (
√ )
√
√
In right
⇒
√
√
Optimum solution for snub cube (Archimedean solid) Application of “HCR’s Theory of Polygon” & “Newton-Raphson Method”
Applications of “HCR’s Theory of Polygon” proposed by Mr H.C. Rajpoot (year-2014) ©All rights reserved
√ √ (
√ )
√
√
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 of 32
congruent equilateral triangular faces at the centre of the
snub cube as follows
(
(√
)
√ (√
)
)
(
√
√
√
√
)
(
√
√
)
(√
)
⇒ (√
)
Similarly, we get the solid angle subtended by each of 6 congruent square faces at the centre of the snub cube
as follow
(
(√
)
√ (√
)
)
Figure 1: An equilateral triangular face with centre M & a square face with centre N having a common vertex A & the
normal distances 𝑯𝑻 𝑯𝒔 respectively from the centre O of the snub cube.
Optimum solution for snub cube (Archimedean solid) Application of “HCR’s Theory of Polygon” & “Newton-Raphson Method”
Applications of “HCR’s Theory of Polygon” proposed by Mr H.C. Rajpoot (year-2014) ©All rights reserved
(
√
√
√
√
)
(√
√
) (√
)
⇒ (√
)
Since a snub cube is a closed surface & we know that the total solid angle, subtended by any closed surface at
any point lying inside it, is (Ste-radian) hence the sum of solid angles subtended by 32 congruent
equilateral triangular & 6 congruent square faces at the centre of the snub cube must be . Thus we have
[ ] [ ]
Hence by setting the values of from eq(III) & (IV) in the above expression we have
* (√
)+ * (√
)+
(√
) (√
)
* (√
) (√
)+
(√
) (√
)
( √
√
) (
√
)
( √ )
( √
√
) (
)
( √
√
) (
)
( ⇒ (
) )
( √
√
) * (
)+
( √
√
) (
)
( √
√
) ( *
+
)
Optimum solution for snub cube (Archimedean solid) Application of “HCR’s Theory of Polygon” & “Newton-Raphson Method”
Applications of “HCR’s Theory of Polygon” proposed by Mr H.C. Rajpoot (year-2014) ©All rights reserved
( √
√
) (
( )
)
( √
√
) [ (
( )
)]
( √
√
) (
( )
)
[
( √
√
)(
( )
) √ ( √
√
)
√ (
( )
)
]
( ( √ √ ) )
√ ( )
( )
⁄
√
(√
( )
)
√ ( ) √
√
(
) (
)(
)
⇒
Now by using Newton-Raphson iteration equation, having second order or quadratic convergence, to
calculated approximate real root of above 8th
degree equation as follows
In order to reduce no. of successive iterations, let’s take the initial approximate root close to the exact root
i.e. which is close to the value of the circumscribed radius of a small rhombicuboctahedron, it is for
the reason that a small rhombicuboctahedron can always be deformed to obtain a snub cube. Thus we can
easily obtain the optimum value of variable by using minimum no. of successive iterations as follows
Iteration-1:
Optimum solution for snub cube (Archimedean solid) Application of “HCR’s Theory of Polygon” & “Newton-Raphson Method”
Applications of “HCR’s Theory of Polygon” proposed by Mr H.C. Rajpoot (year-2014) ©All rights reserved
Iteration-2:
Iteration-3:
Iteration-4:
Iteration-5:
We find that two successive iterations 4 & 5 are giving the same value of real root
hence the optimum value of real root of is
⇒
Normal distance of equilateral triangular faces from the centre of snub cube: The normal
distance of each of 32 congruent equilateral triangular faces from the centre O of a snub cube is given as
√
√
√
⇒ √
It’s clear that all 32 congruent equilateral triangular faces are at an equal normal distance from the
centre of any snub cube.
Solid angle subtended by each of the equilateral triangular faces at the centre of snub cube:
It is given from eq(III) above as follows
(√
) (√
) (√
)
Optimum solution for snub cube (Archimedean solid) Application of “HCR’s Theory of Polygon” & “Newton-Raphson Method”
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 snub cube: The normal distance of each
of 6 congruent square faces from the centre of snub cube is given as
√
√
√
⇒ √
It’s clear that all 6 congruent square faces are at an equal normal distance from the centre of any snub
cube.
Solid angle subtended by each of the square faces at the centre of snub cube: It is given from
eq(IV) above as follows
(√
) (√
) (√
)
(√
)
It’s clear from the above results that the solid angle subtended by each of 6 square faces is greater than the
solid angle subtended by each of 32 equilateral triangular faces at the centre of any snub cube.
It’s also clear from eq(V) & (VII) 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 snub cube i.e. square faces are
the closer to the centre as compared to the equilateral triangular faces in any snub cube.
Important parameters of a snub cube:
1. Inner (inscribed) radius : It is the radius of the largest sphere inscribed (trapped inside) by a
snub cube. The largest inscribed sphere always touches all 6 congruent square faces but does not
touch any of 32 congruent equilateral triangle faces at all since all 6 square faces are closer to the
centre as compared to all 32 triangular faces. Thus, inner radius is always equal to the normal
distance ( ) of square faces from the centre of a snub cube & is given from the eq(VII) as follows
√
Hence, the volume of inscribed sphere is given as
Optimum solution for snub cube (Archimedean solid) Application of “HCR’s Theory of Polygon” & “Newton-Raphson Method”
Applications of “HCR’s Theory of Polygon” proposed by Mr H.C. Rajpoot (year-2014) ©All rights reserved
( √
)
2. Outer (circumscribed) radius : It is the radius of the smallest sphere circumscribing a snub
cube or it’s the radius of a spherical surface passing through all 24 vertices of a snub cube. It is given
as follows
Hence, the volume of circumscribed sphere is given as
3. Surface area : We know that a snub cube has 32 congruent equilateral triangular & 6 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 snub cube with edge length has 32 congruent equilateral triangular
& 6 congruent square faces. Hence, the volume (V) of the snub cube 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 snub cube. It is
calculated as follows
Optimum solution for snub cube (Archimedean solid) Application of “HCR’s Theory of Polygon” & “Newton-Raphson Method”
Applications of “HCR’s Theory of Polygon” proposed by Mr H.C. Rajpoot (year-2014) ©All rights reserved
(
√ √
) ⇒
( √ √
)
( √ √
)
It’s clear from above results that
Construction of a solid snub cube: In order to construct a solid snub cube 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 32 congruent right pyramids with equilateral triangular base of side length & normal height ( )
√
Construct 6 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 three equilateral triangular bases (faces) & a square
base (face) meet at each of 24 vertices. Thus a solid snub cube, with 32 congruent equilateral triangular & 6
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 snub cube. Then, we perform the facing operations on the solid sphere to generate 32
congruent equilateral triangular & 6 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 snub cube 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 snub cube as follows
Now, substituting ⁄ in the above expression, we have
Above relation is very useful for determining the edge length of a snub cube to be produced from a solid
sphere with known diameter D for manufacturing purpose.
Optimum solution for snub cube (Archimedean solid) Application of “HCR’s Theory of Polygon” & “Newton-Raphson Method”
Applications of “HCR’s Theory of Polygon” proposed by Mr H.C. Rajpoot (year-2014) ©All rights reserved
Hence, the maximum volume of snub cube 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 snub cube 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 snub cube of the maximum volume (or with maximum desired edge length)
Conclusions: Let there be any snub cube having 32 congruent equilateral triangular & 6 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 snub cube
Solid angle subtended by each face at the centre of the snub cube
Equilateral triangle
32
√
(√
)
Square
6
√
(√
)
Optimum solution for snub cube (Archimedean solid) Application of “HCR’s Theory of Polygon” & “Newton-Raphson Method”
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
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