Hybrid N-icons
This page deals with Hybrid N-icons. By hybrid, it means it is one half an n-icon and the other half is that of the n-icon's dual. Hybrid n-icons are of even order so N is always even. Like Odd Order N-icons they have at least one discontinuous surface and one discontinuous edge.
They are almost but not quite self dual. If one possesses N vertices it will also have N+1 faces. When the number of facets is low, the differences in the dual are more noticeable. But when the number of facets is high enough the dual looks almost exactly like its base pair. The duals don't have a twist plane and cannot be twisted thus they are not N-icons.
The study of Hybrids N-icons was an additional feature added onto the study. At first I wasn't going to bother with anything more than their curious construction. Then later, I began adding on logic to count and color their continuous surfaces. This led to some interesting discoveries in their patterns which turned out to be challenging to determine.
There is a case with Hybrid N-icons such that when N is a power of 2, there are no twists T such that the N-icon has more than one surface. However, all other Hybrid N-icons do have at least one twist T such that there is more than one surface.
Combinatorial properties of Hybrid N-icons:
- T cannot equal 0. Technically, the twists occur at T = ... -1.5, -0.5, +0.5, 1.5 ... and so on.
- Since the values of T are discreet, for simplicity the twist count is recorded as integers T = ... -2, -1, +1, +2 ...
- Since there is no twist 0, there is no non-chiral base model
- For each N-icon of twist T, it will have a mirror image if a twist of -T is applied, with one exception noted below
- For N, if the facets are uniquely colored, there are N distinct twists ranging from -floor(N/2) through floor(N/2)
- For N, if the facets are all one color then only distinct shapes are considered. Then:
- if (N+2) mod 4 != 0, then there are N distinct twists T ranging from -floor((N+2)/4) through floor((N+2)/4)
- if (N+2) mod 4 = 0, then there are N distinct twists T ranging from -floor((N+2)/4)+1 through floor((N+2)/4)
- in the latter case, an additional form will be non-chiral when abs(T) = (N+2)/4 (no mirror image)
Surface and edge properties of Hybrid N-icons and T lies within the range of distinctive shapes:
- The generalized formula for total surfaces is: (gcd(N, abs(2T-1)) + 1)/2
- They may have no continuous surfaces, CS
- CS will be greater than 0 if one of two cases occurs:
- Case 1: 2T+1 is a factor of 2N, such that 2N mod abs(2T)+1 = 0 then
- CS = abs(T) - 1
- Case 2: is a multiple twist of a Case 1 N-icon where the new T is not a Case 1 N-icon
- CS is derived from the Case 1 N-icon
- They always have 1 discontinuous surface
- The number of continuous edges equals CS + 1
- They always have 1 discontinuous edge
The Surface Count Reflection Property will occur in the following way:
- if N/2 is even:
- Let L = floor((N+2)/4); This is the last distinct twist
- Let C = ceil(L/2)
- if abs(T) > C then T1 = L - abs(T) + 1 then:
- The total number of surfaces at T is the same number as at twist T1 (Surface Count Reflection Property)
Example of Chiral Pairs:
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When N/2 is even, the surface characteristics are completely different than when N/2 is odd. It turns out that there is a reflective nature in respect to T. To determine if there are additional continuous surfaces, we examine if 2N mod abs(2T)-1 = 0. However, this calculation will only equal 0 if the value of T such that L = floor((N+2)/4) and T <= ceil(L/2), which is the half way point between 1 and the maximum number of twists T possible.
Let us use an N-icon of 108 as an example. Let L be the last distinct twist such that L = floor((N+2)/4). For 108, L = 27, so the distinct twist range is between 1 and 27. The half way point, C = ceil(L/2) = 14. So up to 14, the formula 2N mod abs(2T)-1 = 0 will yield valid instances where CS > 0 at 2, 5 and 14. But 23, and 26 are also valid instance where CS > 0 and the forumla will not equal 0. Notice that the second and fifth twist from the beginning is 2 and 5 respectively. Twists 26 and 23 are the second and fifth twists from L (27). In general, for an N-icon Hybrid, if abs(T) > ceil(L/2) then a Hybrid N-icon at T has the same continuous surface properties as one at T = L - abs(T) + 1
N108+T23h and N108+T26h are examples of Case 2 N-icons. The surfaces can be colored based on an earlier twist. N108+T23h can be derived by twisting N108+T5h 18 increments. N108+T26h can be derived by twisting N108+T2h 24 increments.
One additional property of Hybrid N-icons when N/2 is even is that in every case where (N+4) mod 8 = 0, there will always be an instance T at ceil(L/2) which will have CS = abs(T) - 1. For example, N100+T13h has 12 continuous surfaces, for N108+T14h, CS = 13, for N112+T15h, CS = 14, and so on. It is also noteworthy in these cases the twist angle is 45°. In the case where N = 4, (4+4) mod 8 = 0. Then there is such a twist instance T at ceil(4/8) = 1. Then for T = 1, CS = abs(1) - 1 = 0, so N4+T1h has 0 continuous surfaces.
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When N/2 is odd, there is no T reflection phenomona. The first rule for determining which values of N for which there are Hybrid N-icons with continuous surfaces is the same. However determining if 2N mod abs(2T)-1 = 0 can simply be done for every value of T. If the forumla equals 0 then CS = abs(T) - 1.
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There is a unique property of Hybrid N-icons when N/2 is odd. For every such N, when abs(T) = (N+2)/4 the Hybrid N-icons will be non-chiral.
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Credit goes to Adrian Rossiter for the generalized surface formula.
Question or comments about the web page should be directed to polyhedra@bigfoot.com.
The generation of OFF, VRML, and Live3D files was done with Antiprism. The Hedron application by Jim McNeill was used to generate VRML Switch files.
History:
2007-09-06 Initial Release
2007-10-02 Note that Hybrids are not self dual
2007-10-19 Revision: additional language inserted for Case 1 and Case 2 N-icons
2007-11-26 Corrected bracketing on general formula