Side 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 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 N-icons.

A smooth model is dual to the Point Cut N-icon, but in faceted form they are not quite dual to Point Cut N-icons. But even in faceted form, the circuitry of surfaces and edge paths of the two are co-dual.

Combinatorial properties of Side Cut Even Order N-icons:


Surface and edge properties of Side Cut Even Order N-icons and twist T != 0 and T lies within the range of distinctive shapes:


Surface and edge properties of Side Cut Even Order N-icons and twist T = 0:





Example of Chiral Pairs:

8-icon Dual with Twist -1
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8-icon Dual with Twist +1
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An 8-icon Dual, also known as a Octa-Sphericon 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. They are turning in opposite directions.

Each has two discontinuous surfaces and two continuous edges.

Example of a Case 1 to Case 2 transition:

N42+T3s
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N42+T9s
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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 Side Cut Even Order N-icon 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 N-icons 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°.

N4+T0s
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N6+T0s
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N8+T0s
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N16+T0s
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N36+T0s
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Every Side Cut Even Order N-icons 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.

N22+T1s
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N22+T2s
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N22+T3s
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N22+T4s
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N22+T5s
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Here are some Side Cut Even Order N-icons 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.

N12+T2s
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N18+T3s
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N42+T6s
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N42+T7s
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N112+T16s
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If N mod 4 is 0 and T = N/4, the resulting N-icon Dual is not chiral. Here are some of them. Notice the first one, N4+T1, is the Dual of the Sphericon.

N4+T1s
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N8+T2s
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N12+T3s
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N16+T4s
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N32+T8s
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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
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/EvenOrderSideCut

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