Explore the topology of surfaces using the Euler Characteristic.
Polyhedron Parameters
A convex polyhedron satisfies Euler's formula: V - E + F = 2. If you get a different number, the shape might have holes or be non-convex!
Topological Analysis
2
Euler Characteristic (χ)
Genus (g)0
Surface TypeSpherical
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About this calculator
What is Topology?
Topology is a branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing. It is often called 'rubber-sheet geometry'. A key invariant in topology is the Euler Characteristic.
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Pro Tips
For any convex polyhedron (like a cube or pyramid), the Euler Characteristic χ is always 2.
The 'Genus' (g) represents the number of 'holes' in a surface. A sphere has g=0, while a torus has g=1.
Topology treats a coffee mug and a donut as the same shape because both have exactly one hole.
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Fun Facts
"The Seven Bridges of Königsberg is a classic problem that started the field of topology."
"In topology, there is no such thing as distance; distance is a geometric concept, not a topological one."
"A Mobius strip is a famous topological surface with only one side and one edge."
Formula: χ = V - E + F | χ = 2(1 - g) for closed surfaces
