Applied geometry: symmetry within a 3D volume

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Discussion Overview

The discussion revolves around the properties of a 3D shape derived from a cube, specifically focusing on its symmetry and the method of stringing its vertices with yarn. Participants explore the implications of symmetry in the context of geometric shapes, particularly those with high symmetry like the one described, which is hinted to be a tesseract.

Discussion Character

  • Exploratory
  • Conceptual clarification

Main Points Raised

  • One participant describes a 3D shape that inherits properties from a cube and has high symmetry, mentioning that it has 16 vertices and 32 edges.
  • It is noted that this shape allows for all vertices to be connected with a single piece of yarn due to each vertex having an even number of edges, contrasting with a regular cube where vertices have an odd number of edges.
  • The participant expresses a desire for the stringing pattern to maintain symmetry, particularly rotational symmetry, and questions whether there is a scientific basis for achieving this.
  • Another participant requests clarification on the specific shape being discussed.
  • The original poster hints that the shape is a tesseract, providing links for further information and stating that they seek general principles for achieving symmetry in their stringing pattern.

Areas of Agreement / Disagreement

There is no consensus on the specific shape being discussed, as one participant seeks clarification while the original poster hints at a tesseract. The discussion remains exploratory without a definitive resolution on the principles of symmetry in the stringing pattern.

Contextual Notes

The discussion lacks detailed definitions of symmetry and does not resolve the mathematical implications of the stringing pattern. The original poster's approach to symmetry is not fully articulated, leaving open questions about the methods to achieve it.

DaveC426913
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I've got a 3D shape that, um, inherits from a cube (i.e. it retains properties characteristic of cubes). It has high symmetry.

I wish to string all its vertices (of which there are 16) with - I dunno, yarn - forming the edges (of which there are 32). I can do the entire thing with a single piece of yarn (all vertices have an even number of edges, thus no loose ends); AND I can join the loose ends together, making a single loop of yarn. I have physically proven this to be true.

(For comparison, this cannot be done with a regular cube, since all vertices have an odd number of edges, meaning all vertices will need an end of the yarn.)

Now all I want to do is ensure that my pattern of stringing is as symmetrical as possible, i.e. even accounting for the stringing pattern, there should still be at least some symmetry. ISTM, rotational symmetry is the strictest.

Is there a science to this?


Alternately, I could turn this into a teaser and leave you guys to figure it out...

I'll give you a giant hint: its Schläfli symbol is {4,4} (square faces, 4 faces per vertex).
 
Last edited:
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Not one taker?
 
Could you provide a little more information on what shape this is?
 
Well, I was figuring I'd be looking for general principles and do the heavy-lifting myself, but:
http://en.wikipedia.org/wiki/Tesseract"
and http://davesbrain.livejournal.com/287490.html" .

The pattern I've shown has no organization. I want it to be symmetrical.
 
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