GMY-31

Euler characteristic

Polyhedron χ.

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GoverningEuler characteristic

Governing equation

\\chi=V-E+F

where

V: V (—)
E: E (—)
F: F (—)
chi: Euler characteristic (—)

Lecture brief

Historical brief

Pythagoras, Heron, spherical trigonometry and the sphere’s volume are the measuring arts that predate calculus. The sheets compute length, area and angle on plane and sphere. This sheet (GMY-31 — Euler characteristic) is the form associated with Euler characteristic. Working symbols: $V$ , $E$ , $F$ $\rightarrow$ $chi$ . Polyhedron χ. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

chi

from

V

E

F

in Geometry via

\\chi=V-E+F

Polyhedron χ. Use it when a real geometry question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

V = 8.000\,\mathrm{—}

E = 12.000\,\mathrm{—}

F = 6.000\,\mathrm{—}

, the governing relation

\\chi=V-E+F

yields

chi = 2.000\,\mathrm{—}

. One governing identity, SI units, a single sweep on the sheet. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

V V8 —E E12 —F F6 —

Outputs

Euler characteristic chi2.000 —

Reading speed

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Narration of this film

One governing identity, SI units, a single sweep on the sheet.

Polyhedron χ. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube