Point Cut Even Order Nicons
This page deals with Nicons where N is an even number. They are Nicons where a polygon with an even number of edges is swept 180° to generate the base model. An additional restriction is that these Nicons 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 Nicons.
A smooth model is dual to the Side Cut Nicon, but in faceted form they are not quite dual to Side Cut Nicons. But even in faceted form, the circuitry of surfaces and edge paths of the two are codual.
Combinatorial properties of Point Cut Even Order Nicons:
 When T = 0, it will be the nonchiral base model
 For each Nicon 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 nonchiral when abs(T) = N/4 (no mirror image)
Surface and edge properties of Point Cut Even Order Nicons 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 Nicon where the new T is not a Case 1 Nicon
 CS is derived from the Case 1 Nicon
 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 Nicons 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 Nicons of the same N
Example of Chiral Pairs:



A 6icon, also known as a HexaSphericon, 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. 

Example of a Case 1 to Case 2 transition:



There are cases where an Nicon 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 Nicons 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 

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 Nicons 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 Nicon 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:
20090307 Revised commentary on duals
20071126 Corrected bracketing on general formula
20071019 Revision: additional language inserted for Case 1 and Case 2 Nicons