#G09 · Geometry & Trigonometry · Lv.2View in 3D Mind-Map

Euler's Polyhedron Formula & Topology

Leonhard Euler · 1750 CE

V - E + F = 2,\quad \chi = 2 - 2g

\text{Cube: } 8 - 12 + 6 = 2,\quad \text{Icosahedron: } 12 - 30 + 20 = 2

For any convex polyhedron or planar graph on a sphere, Vertices - Edges + Faces = 2—the foundational invariant of topology and 3D mesh geometry.

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Vertices V

Edges E

Faces F

V - E + F = 12 - 30 + 20 = 2 \quad (\text{Sphere } \chi=2 \implies F=20)

Euler Characteristic χ

2

Required Faces F

20 (구면 일치)

E/V Ratio

2.50

Prerequisite Path (11 STEP)

Full Path Analyzer

Click steps in order to follow the 3D neon path from basics to target.

STEP 1#A02Polynomial Expansion & Factoring

(a+b)^2 = a^2 + 2ab + b^2

STEP 2#A03Laws of Exponents & Radicals

a^m \cdot a^n = a^{m+n}

STEP 3#G03Pi, Radian Measure & Circular Sector

l = r\theta

STEP 4#A01Linear Equation & Balance Axiom

ax + b = 0 \implies x = -\frac{b}{a}

STEP 5#G01Pythagorean Theorem

a^2 + b^2 = c^2 \implies c = \sqrt{a^2 + b^2}

STEP 6#A04Quadratic Formula & Discriminant

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

STEP 7#G04Unit Circle Trigonometry & Identities

\cos^2\theta + \sin^2\theta = 1

STEP 8#G07Conic Sections (Parabola, Ellipse, Hyperbola)

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

STEP 9#G022D/3D Euclidean Distance Formula

d(P

STEP 10#G08Spherical Coordinates & Sphere Volume/Area

V = \frac{4}{3}\pi r^3

STEP 11#G09Euler's Polyhedron Formula & Topology

V - E + F = 2

Geometric & Intuitive Insight

다면체에서 면 하나를 떼어내고 평면에 납작하게 펼친 뒤, 삼각형으로 쪼개거나 외곽 모서리를 하나씩 지워나가도 V - E + F 의 값은 절대 변하지 않고 끝내 2로 보존됩니다.

Step-by-Step Derivation

Step 1.1단계: 다면체의 한 면을 제거하여 평면 그래프로 펼치면 면의 수가 F-1 이 됩니다.
Step 2.2단계: 다각형 면에 대각선을 하나 그으면 모서리 E가 1 늘고 면 F도 1 늘어 V - E + F 가 불변입니다.
Step 3.3단계: 외곽 삼각형을 하나씩 제거해 마지막 점 1개(V=1, E=0, F=1 → 바깥면 포함 F=2)에 도달하면 V - E + F = 2

AI & Coding Real-World Use

3D polygon manifold mesh validation in Blender/Unreal and VLSI planar graph routing

#3D Mesh Topology#Graph Theory#CAD Manifold Check

const eulerChar = (V: number, E: number, F: number) => V - E + F;

Geometry & Trigonometry — Related Formulas in This Cluster

Pythagorean Theorem

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Pi, Radian Measure & Circular Sector

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Unit Circle Trigonometry & Identities

Lv.2 · LLM RoPE Encoding

Law of Sines & Law of Cosines

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Trigonometric Addition & Double-Angle Formulas

Lv.3 · Rotation Matrix Proof