Wednesday, January 25, 2017

HP Prime: Volume and Surface Area of Platonic Solids

HP Prime:  Volume and Surface Area of Platonic Solids

About Platonic Solids

A Platonic solid is one of five three dimensional objects consisting of regular polygons which all faces are identical and perfect symmetry is achieved at each vertex (corner).  The five Platonic solids are classified by the number of faces they have:  tetrahedron (4), hexahedron (aka cube) (6), octahedron (8), dodecahedron (12), and icosahedron (20).

Aside from being well known geometric shapes, Platonic solids were considered sacred and thought to play a significant role in cosmology.  In the 360 BC dialogue Timaeus, Plato considered right triangles to be sub-atomic particles, which form the Platonic Solids.  Each of the Platonic solids represents an element:  tetrahedron represents fire, hexahedron represents earth, octahedron represents air, dodecahedron represents ether, and icosahedron represents water.  Those elements help build all of the universe.   Plato also believed that the elements are interchangeable, which particles split up into triangles and rearranging themselves.  [3]

In 1596, Johannes Kepler wrote Mysterium Cosmographicum (The Cosmographic Mystery).  Based on the Copernican system (which considered our Sun as the center of the Universe), Kepler attributed to structure of Solar System with Platonic Solids.  Each planet had its own corresponding sphere where its orbit was located.  Each of the Platonic solids were placed so that they were inscribed and circumscribed by the spheres.  The order went like this:  Mercury, Octahedron, Venus, Icosahedron, Earth, Dodecahedron, Mars, Tetrahedron, Jupiter, Hexahedron, Saturn.  (Uranus and Neptune were not discovered at this time). [4]

Below is some basic geometric and some eccentric information for the Platonic solids:


Platonic Solid
# of Faces
# of Vertices
# of Edges
Volume
Surface Area
Tetrahedron
4
4
6
A^3 * √2 / 12
A^2 * √3
Hexahedron
6
8
12
A^3
A^2 * 6
Octahedron
8
6
12
A^3 * √2 / 3
A^2 * 2 * √3
Dodecahedron
12
20
30
A^3 * (15 + 7 * √5)
A^2 * (3 * √( 20 + 10 *√5))
Icosahedron
20
12
30
A^3 * 5 * (3 + √5) /12
A^2 * 5 * √3
(A = length of a side) 

Platonic Solid
Internal Angle
Element [1]
Philosophy [1]
Chakra [2]
Duals [3]
Tetrahedron
90°
Fire
Balance, Stability
3rd
(none)
Hexahedron
120°
Earth
Earth, Nature
1st
Octahedron
Octahedron
135°
Air
Love, Compassion
4th
Hexahedron
Dodecahedron
150°
Universe/Ether
Spirit, Heavens
5th, 6th, 7th
Icosahedron
Icosahedron
162°
Water
Expression, Creativity
2nd
Dodecahedron


Below are programs to calculate the volume and surface area of each of the Platonic solids

Prime Programs – Platonic Solids

Tetrahedron
Volume:
EXPORT VOLTET(A)
BEGIN
√2*A^3/12;
END;

Surface Area:
EXPORT SURTET(A)
BEGIN
A^2*√3;
END;

Hexahedron:
Volume:
EXPORT VOLHEX(A)
BEGIN
A^3;
END;

Surface Area:
EXPORT SURHEX(A)
BEGIN
6*A^2;
END;

Octahedron:
Volume:
EXPORT VOLOCT(A)
BEGIN
A^3*√2/3;
END;

Surface Area:
EXPORT SUROCT(A)
BEGIN
2*√3*A^2;
END;

Dodecahedron:
Volume:
EXPORT VOLDOD(A)
BEGIN
(15+7*√5)*A^3/4;
END;

Surface Area:
EXPORT SURDOD(A)
BEGIN
(3*√(20+10*√5))*A^2;
END;

Icosahedron
Volume:
EXPORT VOLICO(A)
BEGIN
5*(3+√5)*A^3/12;
END;

Surface Area:
EXPORT SURICO(A)
BEGIN
5*√3*A^2;
END;


Sources

[1]  Punctured Artefact “Symbolism.  The Platonic Solids”  October 13, 2013.  Retrieved January 22, 2017.  Link:  https://puncturedartefact.wordpress.com/2013/10/13/symbolism-the-platonic-solids-tattoo-design-and-culture/

[2] Patinkas.  “The Merkaba, Platonic Solids, & Sacred Geometry” 2014.  Retrieved January 22, 2017.  Link:  http://www.patinkas.co.uk/Merkaba_Feature_Article/merkaba_feature_article.html

[3] Mathpages.  “Platonic Solids and Plato’s Theory of Everything”  Retrieved January 20, 2017.  Link:  http://www.mathpages.com/home/kmath096/kmath096.htm

[4] Wikiepdia.  “Mysterium Cosmographicum”  Retrieved Janaury 23, 2017.  Link:  https://en.wikipedia.org/wiki/Mysterium_Cosmographicum#Shapes_and_the_planets

The first month of 2017 is almost in the books.  Until next time,

Eddie


This blog is property of Edward Shore, 2017.



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