Even Order N-icons Duals
This page deals with N-icons where N is an even number. They are N-icons where a polygon with an even number of edges is swept 180° to generate the base model. An additional restriction is that these N-icons are swept around an axis formed between the midpoint of two opposite edges of the polygon to generate what is called the "Side Cut" base model. Where the polygons are swept around an axis formed between two opposite vertices, the "Point Cut", is covered in Even Order N-icons.
Combinatorial properties of Even Order N-icons Duals:
- When T = 0, it will be the non-chiral base model
- For each N-icon of twist T != 0, 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)+1 through floor(N/2)
- For N, if the facets are all one color then only distinct shapes are considered. Then:
- if N mod 4 != 0, then there are N distinct twists T ranging from -floor(N/4) through floor(N/4)
- if N mod 4 = 0, then there are N distinct twists T ranging from -(N/4)+1 through N/4
- in the latter case, an additional form will be non-chiral when abs(T) = N/4 (no mirror image)
Surface and edge properties of Even Order N-icons Duals and twist T != 0 and T lies within the range of distinctive shapes:
- The generalized formula for total surfaces is: (gcd(N, abs(2T)) + 2)/2
- They can have 0 continuous surfaces, CS
- CS will be greater than 0 if one of two cases occurs:
- Case 1: T is a factor of N, such that N mod abs(T) = 0 then
- if N/T is even then CS = abs(T) - 1
- if N/T is odd then CS = abs(T/2) - 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
- There are always 2 discontinuous surfaces
- The number of continuous edges equals CS + 1
- There are no discontinuous edges
Surface and edge properties of Even Order N-icons Duals and twist T = 0:
- CS = (N/2) - 2
- There are 2 discontinuous surfaces
- The number of continuous edges equals CS + 2
- There are no discontinuous edges
- They will have (N/2) - 2 latitudinal bands and 2 "polar caps"
Example of Chiral Pairs:
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An 8-icon Dual, also known as a Octa-Sphericon Dual, will have a mirror image. Here it is shown as a chiral pair. The first one has a twist of -1 applied and the second one a twist of +1.
Each has two discontinuous surfaces and two continuous edges. |
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Example of a Case 1 to Case 2 transition:
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There are cases where an N-icon of a given surface count can be twisted to another position and have the same surface count. The first occurrence of this for Even Order N-icon Duals is when N28+T2d is twisted four increments to N28+T6d. The latter will have 2 surfaces which is the same as its primary twist position. The first occurrence for 3 surfaces happens twisting N32+T2d four increments to N32+T6d.
Shown to the right is the first occurrence for 4 surfaces. N42+T3d is twisted six increments to become N42+T9d |
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When Twist T = 0, the base model becomes more spherical has N rises. Even Order N-icons Duals tend toward becoming globes with flat north and south polar caps. The half models are also presented showing the polygon which is being swept 90°.
Every Even Order N-icons Dual has at least two discontinuous surfaces. Here, in the case of N = 22, each of the 5 different twists has only 2 surfaces. Note that N22+T2 has a twist of 2 but only has 2 surfaces. This is because 22/2 = 11 which is odd so there are no extra continuous surfaces.
Here are some Even Order N-icons Duals with some continuous surfaces in addition to the two discontinuous surfaces. N12+T2 has a twist of 2 and has 3 surfaces. There are the two discontinuous surfaces plus an additional continuous one. This is because 12/2 = 6 which is even. Therefore the number of additional conitinuous surfaces is abs(T) - 1 which in this case is 1. The same rule applies for N18+T3d. 18/3 = 6 which is even and abs(T) - 1 in this case is equal to 2 for a total of 4 surfaces.
In the case of N42+T6d, 42/6 = 7 which is odd so the number of additional continuous surfaces is abs(T/2) - 1, which is 6/2 - 1 = 2 for a total of 4 surfaces. For N42+T7d, 42/7 = 6 is even, so the number of additional continuous surfaces is abs(T) - 1, or 7 - 1 = 6 for a total of 8 surfaces. For N112+T16d, 112/16 = 7 which is odd so the number of additional continuous surfaces is abs(T/2) - 1, which is 16/2 - 1 = 7 for a total of 9 surfaces.
If N mod 4 is 0 and T = N/4, the resulting N-icon Dual is not chiral. Here are some of them. Notice the first one, N4+T1, is the Dual of the Sphericon.
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
2007-10-19 Revision: additional language inserted for Case 1 and Case 2 N-icons
2007-11-26 Corrected bracketing on general formula
