Symmetries and Transformation Groups of Equilateral Triangle & Icosahedron

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Discussion Overview

The discussion centers on the symmetries and transformation groups of the equilateral triangle and the icosahedron, exploring the number and types of symmetries associated with these geometric shapes. Participants inquire about the mathematical representation of these groups and seek additional resources for understanding symmetries of other geometric objects.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • One participant asks about the number and types of symmetries for the equilateral triangle and the icosahedron.
  • Another participant identifies the groups as |D3| for the equilateral triangle and |A5 x Z2| for the icosahedron.
  • A participant lists 3 reflections and 3 rotations for the equilateral triangle but seeks clarification on the meaning of |D3| and |A5 x Z2|.
  • There is a request for resources to look up symmetries of other geometric objects, including other Platonic solids.
  • A later reply explains that D3 is the dihedral group of order 6 and A5 × Z2 consists of 60 elements from A5 and 2 from Z2, leading to a total of 120 elements for the icosahedron.

Areas of Agreement / Disagreement

Participants generally agree on the identification of the groups and the number of symmetries for the equilateral triangle and the icosahedron, but there is no consensus on the broader implications or additional resources for other geometric objects.

Contextual Notes

Some participants express uncertainty about the definitions and implications of the groups mentioned, and there are unresolved questions regarding how to find symmetries for other geometric shapes.

Who May Find This Useful

This discussion may be useful for those interested in group theory, geometry, and the study of symmetries in mathematical contexts, particularly in relation to geometric objects and their properties.

koolmodee
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How many symmetries (and what symmetries) and how many elements do the transformation groups of the equilateral triangle and the icosahedron have?

thanks
 
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|D3| and |A5 x Z2|, respectively.

How about you start by writing down those of the equilateral triangle. There aren't that many so you can easily list them.
 
3 reflections and 3 rotations, right? But what does |D3| mean?
And what is |A5 x Z2|?

Where can I look up the symmetries for other geometric objects, like for example the other platonic solids?

And what about the number of elements, how do i find out about those?
 
If you've studied groups much, you should recognize what each of those groups are. D3 is the dihedral group of order 6, A5 × Z2 is the direct product of A5, the alternating group of degree 5, which has 5!/2 = 60 elements, and Z2, the cyclic group of order 2.

Thus, the equilateral triangle has 6 symmetries, and the icosahedron has 120.
 

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