Odd Order N-icons
This page deals with N-icons where N is an odd number. They are N-icons where a polygon with an odd number of edges is swept 180° to generate the base model. These N-icons 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 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 (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 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)) + 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 N-icon where the new T is not a Case 1 N-icon
- CS is derived from the Case 1 N-icon
- Note that this means that N-icons derived from prime number sided polygons will always have CS = 0. The only Odd Order N-icons which will have more than one surface will have an non-prime 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 N-icons 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:
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| A 7-icon, also known as a Hepta-Sphericon, 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. They are turning in opposite directions.
Each has one discontinuous surface and one discontinuous edge.
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| When N = 3 it is the simplest N-icon. The 3-icon has been termed the "Conicon" (for being derived from a cone) by Mason Green. It has one discontinuous surface and one discontinuous edge. They are turning in opposite directions.
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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 Odd Order N-icons 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
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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 N-icons with more than one surface. In order for an Odd Order N-icon 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 N-icons derived from prime number sided polygons will always have 0 continuous surfaces. The only Odd Order N-icons 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).
Question or comments about the web page should be directed to PolyhedraSmith@gmail.com.
The generation of OFF and VRML files was done with
Antiprism. The
Hedron application by
Jim McNeill was used to generate VRML Switch files.
History:
2024-06-17 Use base.css
2024-06-13 Switch to Multi OFF Viewer
2023-09-01 Switch to Simple OFF Viewer and X_Ite VRML Viewer
2023-03-01 Open Interactive Viewer from model gifs
2019-03-12 Changed email address from defunct bigfoot.com
2019-03-07 Switch from Live3D to OFF viewer
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
2007-09-06 Initial Release
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Link to this page as http://www.interocitors.com/polyhedra/n_icons/OddOrder
