Side 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 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 Point Cut Even Order Nicons.
A smooth model is dual to the Point Cut Nicon, but in faceted form they are not quite dual to Point Cut Nicons. But even in faceted form, the circuitry of surfaces and edge paths of the two are codual.
Combinatorial properties of Side 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 Side 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)/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 Nicon where the new T is not a Case 1 Nicon
 CS is derived from the Case 1 Nicon
 There are always 2 discontinuous surfaces
 The number of continuous edges equals CS + 1
 There are no discontinuous edges
Surface and edge properties of Side Cut Even Order Nicons 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"
 They are the dual of Point Cut Even Order Nicons of the same N
Example of Chiral Pairs:



An 8icon Dual, also known as a OctaSphericon 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. 

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 Side Cut Even Order Nicon is when N28+T2s is twisted four increments to N28+T6s. The latter will have 2 surfaces which is the same as its primary twist position. The first occurrence for 3 surfaces happens twisting N32+T2s four increments to N32+T6s.
Shown to the right is the first occurrence for 4 surfaces. N42+T3s is twisted six increments to become N42+T9s 

When Twist T = 0, the base model becomes more spherical has N rises. Side Cut Even Order Nicons 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 Side Cut Even Order Nicons 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 Side Cut Even Order Nicons 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+T3s. 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+T6s, 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+T7s, 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+T16s, 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 Nicon 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:
20090307 Revised commentary on duals
20071126 Corrected bracketing on general formula
20071019 Revision: additional language inserted for Case 1 and Case 2 Nicons
20070906 Initial Release