Symmetries and Transformation Groups of Equilateral Triangle & Icosahedron

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SUMMARY

The discussion focuses on the symmetries and transformation groups of the equilateral triangle and the icosahedron. The equilateral triangle has 6 symmetries represented by the dihedral group |D3|, which includes 3 reflections and 3 rotations. In contrast, the icosahedron has 120 symmetries, represented by the group |A5 x Z2|, where A5 is the alternating group of degree 5 with 60 elements, and Z2 is the cyclic group of order 2. Understanding these groups is essential for studying geometric transformations.

PREREQUISITES
  • Understanding of group theory concepts, specifically dihedral and alternating groups.
  • Familiarity with geometric transformations, including reflections and rotations.
  • Knowledge of the properties of Platonic solids.
  • Basic mathematical notation and terminology related to group orders.
NEXT STEPS
  • Research the properties of dihedral groups, specifically |D3| and its applications.
  • Explore the structure and significance of the alternating group A5.
  • Study the symmetries of other Platonic solids and their corresponding transformation groups.
  • Learn about the direct product of groups and its implications in group theory.
USEFUL FOR

Mathematicians, students of abstract algebra, and anyone interested in geometric transformations and symmetry analysis in mathematical structures.

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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