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.
