This page deals with Hybrid Nicons. By hybrid, it means when half of a Point Cut Even Order Nicon is joined with a Side Cut of the same N. Hybrid nicons are of even order so N is always even. Like Odd Order Nicons they have at least one discontinuous surface and one discontinuous edge.
A smooth model would be self dual, but in faceted form they are not quite self dual. But even in faceted form, the circuitry of surfaces and edge paths is self dual.
The study of Hybrids Nicons 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 Nicons such that when N is a power of 2, there are no twists T such that the Nicon has more than one surface. However, all other Hybrid Nicons do have at least one twist T such that there is more than one surface.
Combinatorial properties of Hybrid Nicons:
Surface and edge properties of Hybrid Nicons and T lies within the range of distinctive shapes:
The Surface Count Reflection Property will occur in the following way:
Example of Chiral Pairs:



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 Nicon 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 Nicon Hybrid, if abs(T) > ceil(L/2) then a Hybrid Nicon 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 Nicons. 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 Nicons 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.





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 Nicons 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.


There is a unique property of Hybrid Nicons when N/2 is odd. For every such N, when abs(T) = (N+2)/4 the Hybrid Nicons will be nonchiral.


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:
20090307 Revised commentary on duals
20071126 Corrected bracketing on general formula
20071019 Revision: additional language inserted for Case 1 and Case 2 Nicons
20071002 Note that Hybrids are not self dual
20070906 Initial Release