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Technical notes on Hub and Rod Construction Toy Systems:

A test for whether or not a Hub and Rod system follows the canonical format is whether or not you can construct a logarithmic spiral with it:

• The first triangle that a tinkertoy type system (TTS) can make uses three hubs (in fact each triangle uses three hubs), two short rods and one long rod.
  • Use the two short rods to connect the three hubs together so as to form a right angle.
  • The long rod is used as the hypotenuse to finish off the triangle.
  • The next triangle in the spiral uses the hypotenuse of the previous triangle as one of its legs.
    • So in this case leg length = long rod and, as it turns out, hypotenuse length = 3 * short rod.

We will call the rods that form the right angle 'legs' and the rods that form the hypotenuse 'hyp.'
The following table shows the relationship:

with the entries under
• 'a' being the short rods
and those under
• 'b' being the long rods:

Table 3:
Number of rods per connection in
the logarithmic-spiral triangles.
Leg:     |Hyp:         | Log Triangle
a     b  |  a     b    | Li
1     0  |  0     1    | L1
0     1  |  3     0    | L2
3     0  |  1     2    | L3
1     2  |  7     0    | L4
7     0  |  3     4    | L5
3     4  | 15     0    | L6
..   ..  | ..    ..    | ..
________________________________

In equation form - For i even:
leg a_i = 2^i/2 - 1
leg b_i = 2^(i/2)-1
hyp a_i = 2^(i/2)+1 - 1
hyp b_i = 0
We leave i odd as an exercise to the reader (hint: inspect the table below)

Table 4:
Number of rods per connection in
the plane-tiling triangles.
Leg:     | Hyp:        | Tiling Triangle:
a     b  |   a     b   | Tn
1     0  |   0     1   | T1
3     0  |   1     2   | T2
5     0  |   2     3   | T3
7     0  |   3     4   | T4
Formula:
2n-1  0  |  n-1    n   | Tn

Notice that if rods cannot be directly connected together then rods of the above lengths need to be included in the set. For example, for L5 the rod length for the legs is equal in length to seven of the short rods. And the rod length for the hypotenuse is equal in length to three of the short rods plus four of the long ones. Or, if you have connectors that look like short rods then, you can do the hypotenuse by tying the four long rods together with three connectors (i.e., long rod, connector, long rod, connector, long rod, connector, long rod). And the legs can be constructed with four short rods and three connectors.

• Tinkertoy
• K'nex and
• Xox

all pass the logarithmic spiral test - but only

• Xox passes it intrinsically.

Below is the beginning of a logarithmic spiral - L1 and L2 - constructed with SystemXox 3000 parts. Note how the hypoteneuse of the second trianlge is constructed from three short rods. The hubs and short rods are gold and the long rods are red.

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