The 5 Platonic Solids
On Wednesday, I went with my friends to the Exploratorium, and Dad boutght us each a gift. I chose a “Plato’s Glo- Mobile” kit, where you create 5 polyhedrons, the Platonic Solids, out of sticks and soft, glow in the dark connectors. It was very fun! Here’s the story about it:
Hundreds of years ago, Plato produced the five Platonic Solids, which all were polyhedrons. There was Tetrahedron, which resembled a triangular pyramid. There was also Octahedron, the eight-triangular-sided polyhedron. Then, there was Icosahedron, which contained of twenty triangular sides. Hexahedron resembled a cube, and Dodecahedron consisted of twelve pentagonal sides.
He also associated them to five different “atomsâ€. Since Tetrahedron has the least sides, Plato gave it fire, the lightest “element†of that time. Additionally, it had the sharpest sides, so he thought it was responsible of the sharp pain. Octahedron had the second least sides, so he granted it the second heaviest atom, air. Icosahedron was the last one that had equilateral triangle sides, so he gave it water. He thought the Hexahedron was stable, and the Earth was stable too, so he gave it Earth. He thought Dodecahedron was very peculiar, so he gave it the Cosmos.
Tetrahedron
|
Fire
-60 sticks
|
Please note: when adding levels to the Tetrahedron as above, it has an interesting mathematical property to count the number of connectors and sticks. For example, when it is one level Tetrahedron, it has 4 connectors and 6 sticks; when it is two level, it has 10 connectors and 24 sticks. So if you keep adding the level, how many number connectors and sticks for you to construct 5 level Tetrahedron? how about 10 level one? (Answer in the comments of this post)
Octahedron
|
Air
-50 sticks
|
Icosahedron
|
Water
-30 sticks
|
Dodecahedron
|
Cosmos
-90 sticks
|
Hexahedron
COMING SOON!
X-Ray Yuhua said,
April 9, 2007 @ 8:57 pm
The formula for counting the number of connectors and sticks to construct the Tetrahedron with different levels:
c(n): the number of connectors at level n;
C(n): the sum of all number of connectors up to level n,
i.e., C(n) = c(n) + c(n-1) + … + c(1) + c(0)
S(n): the number of sticks to construct n-level Tetrahedron.
Let’s observe the easy ones first:
n=0: c(0)=1, C(0)=1, S(0)=0;
n=1: c(1)=3, C(1)=c(1)+c(0)=4, S(1)=6
n=2: c(2)=6, C(2)=c(2)+c(1)+c(0)=10, S(2)=24
the pattern of c(n) can be formulated as:
c(n) = c(n-1) + (n+1)
the patter of C(n) can be formulated as:
C(n) = C(n-1) + c(n)
the pattern of S(n) can be formulated as:
S(n) = C(n-1) * 6
According to above three formula, here are the answers for level 1 to level 10:
So the answer for a 5 level Tetrahedron will require 56 connectors and 210 sticks, and 10 level one will require 286 connectors and 1320 sticks if you ever build such a gigantic Tetrahedron.