| 6-icon Hybrid with Twist -1 |
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| 6-icon Hybrid with Twist +1 |
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Each has one discontinuous surface and one discontinuous edge.
This page deals with Hybrid N-icons. By hybrid, it means when half of a Point Cut Even Order N-icon is joined with a Side Cut of the same N. 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.
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 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:
Surface and edge properties of Hybrid N-icons and T lies within the range of distinctive shapes:
The Surface Count Reflection Property will occur in the following way:
Example of Chiral Pairs:
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| A 6-icon Hybrid will have a mirror image. Here it is shown as a chiral pair. They are self dual. The first one has a twist of -1 applied and the second one a twist of +1. They are turning in opposite directions. Each has one discontinuous surface and one discontinuous edge. |
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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Question or comments about the web page should be directed to PolyhedraSmith@gmail.com.
The generation of OFF and VRML files was done with Antiprism. The Hedron application by Jim McNeill was used to generate VRML Switch files.
History:
2024-06-17 Use base.css
2024-06-13 Switch to Multi OFF Viewer
2023-09-01 Switch to Simple OFF Viewer and X_Ite VRML Viewer
2023-03-01 Open Interactive Viewer from model gifs
2019-03-12 Changed email address from defunct bigfoot.com
2019-03-07 Switch from Live3D to OFF viewer
2009-03-07 Revised commentary on duals
2007-11-26 Corrected bracketing on general formula
2007-10-19 Revision: additional language inserted for Case 1 and Case 2 N-icons
2007-10-02 Note that Hybrids are not self dual
2007-09-06 Initial Release
