Odd Order Nicons
This page deals with Nicons where N is an odd number. They are Nicons where a polygon with an odd number of edges is swept 180° to generate the base model. These Nicons are swept around an axis formed between one vertex to the midpoint of the edge opposite that vertex.
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.
Combinatorial properties of Odd 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 (always chiral)
 For N, if the facets are uniquely colored, there are N distinct twists ranging from floor(N/2) through floor(N/2)
 For N, if the facets are all one color the same statistics apply
Surface and edge properties of Odd 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)) + 1)/2
 They may have no 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
 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
 Note that this means that Nicons derived from prime number sided polygons will always have CS = 0. The only Odd Order Nicons which will have more than one surface will have an nonprime N and a twist abs(T) > 1
 They always have 1 discontinuous surface
 The number of continuous edges equals CS
 They always have 1 discontinuous edge
Surface and edge properties of Odd Order Nicons and twist T = 0:
 CS = floor(N/2)
 There is 1 discontinuous surface
 The number of continuous edges equals CS
 There is 1 discontinuous edge
 They will have floor(N/2) latitudinal bands and 1 "polar cap"
 They are the self dual
Example of Chiral Pairs:



A 7icon, also known as a HeptaSphericon, will have a mirror image. Here it is shown as a chiral pair. The first one has a twist of 2 applied and the second one a twist of +2.
Each has one discontinuous surface and one discontinuous edge. 




When N = 3 it is the simplest Nicon. The 3icon has been termed the "Conicon" (for being derived from a cone) by Mason Green. It has one discontinuous surface and one discontinuous edge. 

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 Odd Order Nicons is when N15+T2 is twisted four increments to N15+T6. The latter will have 2 surfaces which is the same as its primary twist position. The first occurrence for 3 surfaces happens twisting N35+T5 five increments to N35+T10.
Shown to the right is the first occurrence for 4 surfaces. N35+T7 is twisted seven increments to become N35+T14 

When Twist T = 0, the base model becomes more spherical has N rises. They can be thought of as globes with one polar cap. A curious thing about these "globes" is that no matter how many longitudes they are divided into, the number of faces equals the number of vertices.
The half models are also presented showing the polygon which is being swept 90°. When N = 3, an equalateral triangle is swept 180° to form a cone.
Here are some Odd Order Nicons with more than one surface. In order for an Odd Order Nicon to have more than the one default discontinuous surface, it must be of the type N+T where N mod T = 0. Note that this means that Nicons derived from prime number sided polygons will always have 0 continuous surfaces. The only Odd Order Nicons which will have more than one surface will have twist T > 0 which is a factor of N. The number of additional continuous surfaces will be floor(T/2).
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