Triakis Icosahedron Calculator

Calculator and formulas for calculating a triakis icosahedron

The Icosahedral Triangle Master - 60 Triangular Faces in Golden Perfection!

Triakis Icosahedron Calculator

The Triakis Icosahedron

A Triakis Icosahedron is a Catalan solid with 60 isosceles triangular faces - dual to the truncated dodecahedron.

Enter Known Parameter
Which parameter is known?
Value of the selected parameter
Triakis Icosahedron Calculation Results
Long Edge a:
Short Edge b:
Surface Area A:
Volume V:
Circumradius RK:
Inradius RI:
Triakis Icosahedron Properties

The Icosahedral Triangle Master: Dual to the truncated dodecahedron

60 Triangular Faces 90 Edges 32 Vertices Golden Ratio

Triakis Icosahedron Structure

Triakis Icosahedron

The icosahedral triangle master with 60 faces.
Dual to the truncated dodecahedron.

What is a Triakis Icosahedron?

A Triakis Icosahedron is a magnificent golden ratio-based Catalan solid:

  • Definition: Polyhedron with 60 isosceles triangular faces
  • Faces: Each face is an isosceles triangle with two different edge types
  • Dual: To the truncated dodecahedron
  • Vertices: 32 vertices with different coordination
  • Edges: 90 edges in two different lengths
  • Symmetry: Icosahedral symmetry group

Geometric Properties of the Triakis Icosahedron

The Triakis Icosahedron exhibits beautiful icosahedral geometric properties:

Basic Parameters
  • Edge Types: Long edges (a) and short edges (b)
  • Faces: 60 congruent isosceles triangles
  • Euler Characteristic: V - E + F = 32 - 90 + 60 = 2
  • Dual Form: Truncated dodecahedron
Icosahedral Properties
  • Catalan Solid: Dual to Archimedean solid
  • Triangular Faces: Each face is an isosceles triangle
  • Golden Ratio: Proportions contain φ and √5
  • Icosahedral Symmetry: 120 symmetry operations

Mathematical Relationships

The Triakis Icosahedron follows complex golden ratio mathematical laws:

Volume Formula
V ≈ 2.35·a³

Complex formula with golden ratio. Icosahedral in nature.

Surface Area Formula
A ≈ 8.83·a²

Sum of 60 isosceles triangles. Golden ratio geometry.

Applications of the Triakis Icosahedron

Triakis Icosahedra find applications in icosahedral studies:

Scientific Research
  • Icosahedral crystallography
  • Viral capsid structures
  • Golden ratio studies
  • Mathematical symmetry
Engineering
  • Icosahedral structures
  • Golden ratio design
  • Geometric optimization
  • Mathematical modeling
Education
  • Golden ratio education
  • Icosahedral symmetry studies
  • Advanced geometry courses
  • Mathematical beauty demonstrations
Art & Design
  • Icosahedral sculptures
  • Golden ratio art installations
  • Mathematical decorations
  • Geometric pattern design

Formulas for the Triakis Icosahedron

Long Edge (a)
\[a = \frac{22b}{15-\sqrt{5}}\] \[a \approx 1.724 \cdot b\]

Long edge in terms of short edge with √5

Short Edge (b)
\[b = \frac{a(15-\sqrt{5})}{22}\] \[b \approx 0.58 \cdot a\]

Short edge relationship with golden ratio

Surface Area (A)
\[A = \frac{15a^2\sqrt{109-30\sqrt{5}}}{11}\] \[A \approx 8.829 \cdot a^2\]

Surface area with complex √5 dependencies

Volume (V)
\[V = \frac{5a^3(5+7\sqrt{5})}{44}\] \[V \approx 2.347 \cdot a^3\]

Volume with golden ratio coefficients

Circumradius (RK)
\[R_K = \frac{a(1+\sqrt{5})}{4}\] \[R_K \approx 0.809 \cdot a\]

Circumradius with golden ratio factor

Inradius (RI)
\[R_I = \frac{a}{4}\sqrt{\frac{10(33+13\sqrt{5})}{61}}\] \[R_I \approx 0.797 \cdot a\]

Inradius with complex √5 dependency

Isosceles Triangle Properties
60 Triangles
All isosceles and congruent
Two Edge Types
Long (a) and short (b) edges
Golden Ratio
All proportions contain √5

Each triangle has one long edge (a) and two short edges (b) in golden ratio proportions

Calculation Example for a Triakis Icosahedron

Given
Long Edge a = 10

Find: All properties of the icosahedral triangle master

1. Surface Area Calculation
\[A = 8.829 \times 10^2\] \[A = 8.829 \times 100\] \[A = 882.9\]

The surface area is 882.9 square units

2. Volume Calculation
\[V = 2.347 \times 10^3\] \[V = 2.347 \times 1000\] \[V = 2347\]

The volume is 2347 cubic units

3. Short Edge
\[b = 0.58 \times 10 = 5.8\]

The short edge is 5.8 units

4. Radii
\[R_K = 0.809 \times 10 = 8.09\] \[R_I = 0.797 \times 10 = 7.97\]

Circumradius: 8.09, Inradius: 7.97 units

5. The Icosahedral Triangle Master
Long Edge a = 10.0 Short Edge b = 5.8 Surface Area A = 882.9
Volume V = 2347 Circumradius RK = 8.09 Inradius RI = 7.97
60 Triangular Faces ⭐ Golden Ratio 🔳 Icosahedral

The icosahedral triangle master with perfect golden ratio symmetry

The Triakis Icosahedron: The Icosahedral Triangle Master

The Triakis Icosahedron stands as one of the most mathematically sophisticated and geometrically elegant among the Catalan solids, representing the perfect fusion of icosahedral symmetry and golden ratio mathematics. With its 60 congruent isosceles triangular faces, each having two different edge lengths in golden ratio proportions, it demonstrates how the fundamental mathematical constant φ = (1+√5)/2 can organize three-dimensional space into forms of extraordinary complexity and beauty. As the dual to the truncated dodecahedron, it transforms the mixed face arrangements of its Archimedean partner into a uniform pattern of triangular faces, creating a polyhedron that serves as both a mathematical masterpiece and a natural expression of icosahedral perfection.

The Icosahedral Foundation

The Triakis Icosahedron showcases icosahedral golden ratio mathematics:

  • 60 Isosceles Triangles: Each face has one long edge and two short edges
  • Icosahedral Symmetry: 120 symmetry operations of the highest three-dimensional order
  • Golden Ratio Proportions: All relationships involve φ and √5
  • Edge Diversity: Two different edge lengths in golden ratio
  • Complex Mathematics: Formulas with nested √5 terms
  • Icosahedral Heritage: Related to dodecahedron and icosahedron
  • Natural Perfection: Appears in biological and crystalline structures

Catalan Heritage and Icosahedral Duality

Catalan Excellence

As one of the most complex Catalan solids, the Triakis Icosahedron perfectly demonstrates the principle of duality with the truncated dodecahedron, showing how mixed face arrangements become uniform triangular patterns.

Icosahedral Duality

The duality with the truncated dodecahedron creates a fascinating relationship where the complex face types (triangles and decagons) of the dual become uniform isosceles triangles with consistent proportions.

Triangular Perfection

With 60 faces arranged in perfect icosahedral symmetry, each triangle maintains golden ratio edge relationships, creating a form where mathematical perfection and geometric beauty converge.

Mathematical Beauty

The formulas demonstrate the elegant way golden ratio mathematics can describe complex three-dimensional forms, with √5 appearing naturally in all calculations and edge relationships.

The Golden Ratio in Triangular Form

The Triakis Icosahedron is a masterpiece of φ-based geometry:

Edge Relationships

The ratio between long and short edges follows golden ratio mathematics: a ≈ 1.724·b, derived from the relationship 22/(15-√5), showcasing how φ naturally appears in icosahedral constructions.

√5 Mathematics

All formulas contain √5 terms, directly connecting to the golden ratio φ = (1+√5)/2. The complex surface area and volume formulas show how √5 governs the polyhedron's properties.

Triangular Harmony

Each of the 60 isosceles triangles has identical proportions determined by golden ratio relationships, creating a perfect balance between geometric complexity and mathematical elegance.

Icosahedral Harmony

The icosahedral symmetry and golden ratio relationships make this polyhedron appear in nature and crystalline structures, demonstrating deep natural mathematical principles.

Scientific and Natural Significance

The Triakis Icosahedron appears throughout advanced science:

  • Icosahedral Biology: Related to viral capsid structures and biological forms
  • Crystallography: Model for icosahedral crystal structures and quasicrystals
  • Golden Ratio Studies: Perfect model for φ-based three-dimensional forms
  • Mathematical Research: Study object for icosahedral symmetry and duality
  • Fullerene Chemistry: Related to complex carbon structures
  • Art and Architecture: Inspiration for golden ratio-based design
  • Educational Excellence: Advanced demonstration of Catalan solid principles

Construction and Golden Precision

Golden Precision

Manufacturing requires precise implementation of golden ratio relationships in each triangular face. The edge ratios must maintain exact proportions to preserve the icosahedral symmetry.

Mathematical Complexity

The complex √5 terms in formulas require high-precision arithmetic, but the results create one of the most mathematically beautiful polyhedra in existence.

Natural Inspiration

The appearance in biological structures provides a natural template for understanding how golden ratio mathematics manifests in living systems and natural forms.

Aesthetic Excellence

The golden ratio proportions in every face create inherent aesthetic appeal, making physical models particularly beautiful and suitable for artistic and educational display.

Cultural and Educational Impact

Golden Ratio Education

As a complex manifestation of golden ratio mathematics, it provides an excellent teaching tool for understanding how φ creates harmony in advanced geometric forms.

Symmetry Studies

The icosahedral symmetry and dual edge types make it ideal for studying the most complex three-dimensional symmetry groups and their mathematical properties.

Artistic Inspiration

Artists and designers frequently use this form for its inherent golden ratio beauty, creating sculptures and installations that embody mathematical harmony in physical space.

Scientific Wonder

Its appearance in biological structures demonstrates the profound connection between mathematical beauty and natural efficiency, inspiring research into the fundamental role of geometry in life.

Summary

The Triakis Icosahedron stands as a perfect example of how the golden ratio and icosahedral symmetry can combine to create forms of extraordinary mathematical elegance and natural beauty. Its 60 isosceles triangular faces, each with edges in golden ratio proportions and arranged in perfect icosahedral symmetry, make it both a mathematical masterpiece and a natural wonder. From its role in biological structures to its applications in advanced crystallography and symmetry studies, it demonstrates the deep connections between mathematical beauty, natural phenomena, and aesthetic perfection. As a bridge between pure mathematics and biological reality, the Triakis Icosahedron continues to inspire researchers, educators, artists, and anyone fascinated by the golden ratio's power to create perfect harmony in the most complex three-dimensional forms.

Disdyakis Triacontahedron • Deltoidal Hexecontahedron • Deltoidal Icositetrahedron • Hexakis Octahedron • Pentagonal Icositetrahedron • Pentakis Dodecahedron • Rhombic Dodecahedron • Rhombic Triacontahedron • Tetrakis Hexahedron • Triakis Octahedron • Triakis Tetrahedron • Triakis Icosahedron