Peter Saveliev

Dr. Saveliev is a professor of mathematics at Marshall University, Huntington WV, USA. He has been involved in research in algebraic topology and several other fields. The non-academic projects have been: digital image analysis, automated fingerprint identification, and image matching for missile navigation/guidance. He holds a mathematics Ph.D. from the University of Illinois at Urbana-Champaign.

• Topology Illustrated, book 2016
• Areas of interest:
  • Discrete differential geometry and calculus
  • Algebraic topology
  • Image and data analysis
• Publications
• Current classes

Once upon a time I took a better look at the poster of Drawing Hands by Escher hanging in my office and realized that what is depicted isn't symmetric! I decided to fix the problem and I call the result Painting Hands.

The picture illustrates how the antipodal map (aka the central symmetry) reverses orientation in the odd dimensions and preserves it in the even dimensions. That's why to be symmetric the original would have to have two right hands!

More: This symmetry is supposed to be an involution of the $3$-space; therefore its diagonalized matrix has only $\pm 1$ on the diagonal. With an even number of $-1$s you have two right (or two left) hands. With an odd number of $-1$s you have a right and a left hands. However, a single $-1$ gives us a mirror symmetry: pen draws pen. We are left with a single option: three $-1$s, i.e., the central symmetry.

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Note: I am frequently asked, what should "Saveliev" sound like? I used to care about that but got over that years ago. So, one can just read it as is; "Sav-leeeeeev" seems the most popular. Or, call me Peter.