The DNA of a polyhedron

It’s rather amazing that the characteristics of a polyhedron, which is a topological entity, can all be determined by merely its basic constituent unit – it’s almost like the DNA of the entity. No matter what the entity, or complex (in the language of combinatorial topology), is, as long as you can reduce it to its basic constituent unit, or cell, you would know everything you can know about it and its family members.

The work goes into the way one reduces the complex in order to accurately obtain its cell. The primary knowledge that one needs to possess is the characteristics of the different cells that can make up complexes. Now, in combinatorial topology, naturally, the characteristics that we can label and measure are things that we can count. For example, the characteristics of a triangle can be described by its number of faces, or its number of edges, or its number of vertices. What then is the combination that we can make from these numbers that will allow us to reduce a complex in a consistent way to its cell? The answer was found via the genius of Euler, one of the most versatile and prolific mathematicians of the 18th century – Euler’s formula for polyhedra – the number of faces minus the number of edges plus the number of vertices.

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~ by musafiremes on June 20, 2007.

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