Point Cut Even Order N-icons
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 two opposite vertices of the polygon to generate what is called the "Point Cut" base model. Where the polygons are swept around an axis formed between the midpoint of two opposite edges, the "Side Cut", is covered in Side Cut Even Order N-icons.
A smooth model is dual to the Side Cut N-icon, but in faceted form they are not quite dual to Side Cut N-icons. But even in faceted form, the circuitry of surfaces and edge paths of the two are co-dual.
Combinatorial properties of Point Cut Even Order N-icons:
- 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 Point Cut Even Order N-icons 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
- They have at least 1 continuous surface, CS
- CS will be greater than 1 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)
- if N/T is odd then CS = abs(T/2)
- 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 no discontinuous surfaces
- The number of continuous edges equals CS - 1
- They always have 2 discontinuous edges
Surface and edge properties of Point Cut Even Order N-icons and twist T = 0:
- CS = N/2
- There are no discontinuous surfaces
- The number of continuous edges equals CS - 1
- There are no discontinuous edges
- They will have N/2 latitudinal bands
- In polyhedral form they are the dual of Side Cut Even Order N-icons of the same N
Example of Chiral Pairs:
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A 6-icon, also known as a Hexa-Sphericon, 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 one continuous surface and two discontinuous 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 Point Cut Even Order N-icons is when N28+T2p is twisted four increments to N28+T6p. The latter will have 2 surfaces which is the same as its primary twist position. The first occurrence for 3 surfaces happens twisting N42+T3p six increments to N42+T9p.
Shown to the right is the first occurrence for 4 surfaces. N56+T4p is twisted eight increments to become N56+T12p |
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When Twist T = 0, the base model becomes more spherical has N rises. The half models are also presented showing the polygon which is being swept 90°. Notice N36+T0p is like a globe with 10 degree latitudinal gradients, and also faceted at 10 degree longitudinal gradients.
Here are some Point Cut Even Order N-icons with more than one surface. Note that N28+T4p has a twist of 4 but only has 2 surfaces. This is because 28/4 = 7 which is odd. Therefore the number of surfaces is abs(T/2) or, in this case, 2. The same rule applies for N42+T6p. 42/6 = 7 which is odd and T/2 in this case is equal to 3. The other three examples all have an N/T which is an even number so the number of surfaces is equal to abs(T).
If N mod 4 is 0 and T = N/4, the resulting N-icon is not chiral. Here are some of them. Notice the first one, N4+T1p, is 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:
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
