The Amount of Joints and Rods in Various Different Structures
I intend to draw the structures I am investigating on isometric grid
paper and record the results in tables. I will then state formulae to
calculate the amount of rods or joints in any size structure. To back
up my formulae I will be making predictions and proving them to be
correct (hopefully!)
Joints
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Inside the structures I will be investigating the joints I predict to
find are the following:
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three joint-3J four joint-4J five joint-5J six joint-6J
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Found on the Found on the Found on the Found inside
vertices edges faces
I will now investigate the joints above and any others I find in cubes
and cuboids.
Joints In Cubes
===============
Cube Size
3J
4J
5J
6J
TJ
2x2x2
8
12
6
1
27
3x3x3
8
24
24
8
64
4x4x4
8
36
54
27
125
5x5x5
8
48
96
64
216
6x6x6
8
60
150
125
343
7x7x7
8
72
216
216
512
8x8x8
8
84
294
343
729
I will now explain how I collected the results above and state
algebraic formulae to calculate the amount of each type of joint in
any size cube.
In reference to algebra the letter n=any length minus one
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3J
==
As predicted these joints were only found on the vertices of the
structures. Because all of the structures I investigated were the same
they all had the same amount of vertices.
I will first start to count the number of small cubes out of the 27;
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