Pentagonal Bipyramid Calculator
Calculator and formulas for a pentagonal bipyramid (Johnson solid J13)
Pentagonal Bipyramid Calculator
The Pentagonal Bipyramid
The pentagonal bipyramid is a Johnson solid (J13) consisting of 10 equilateral triangles.
Johnson Solid J13 Properties
The pentagonal double pyramid: Two pentagonal pyramids joined at their bases
Pentagonal Bipyramid Structure
The pentagonal double pyramid with golden ratio beauty.
Johnson solid J13.
What is a pentagonal bipyramid?
The pentagonal bipyramid is a fascinating Johnson solid:
- Definition: Two pentagonal pyramids joined at their pentagonal bases
- Johnson solid: J13 in the classification
- Faces: 10 congruent equilateral triangles
- Vertices: 7 vertices total
- Edges: 15 edges (5+5+5)
- Symmetry: Perfect D5h symmetry
Geometric Properties of the Pentagonal Bipyramid
The pentagonal bipyramid shows remarkable geometric properties:
Basic Parameters
- Faces: 10 equilateral triangles
- Vertices: 7 vertices (5 equatorial, 2 polar)
- Edges: 15 edges (all equal length)
- Euler characteristic: V - E + F = 7 - 15 + 10 = 2
Special Properties
- Deltahedron: All faces are triangles
- Equilateral: All triangles are equilateral
- Golden ratio: Connected to pentagonal geometry
- Convex: No concave edges or vertices
Mathematical Relationships
The pentagonal bipyramid follows elegant mathematical laws:
Volume Formula
Contains the golden ratio √5. Beautiful and elegant.
Surface Area Formula
10 equilateral triangles. Simple √3 relationship.
Applications of the Pentagonal Bipyramid
Pentagonal bipyramids find applications in various fields:
Architecture & Construction
- Pentagonal dome structures
- Decorative spires
- Structural supports
- Architectural elements
Science & Technology
- Crystallographic structures
- Molecular geometry
- Optical components
- Mechanical parts
Education & Teaching
- Geometry instruction
- 3D geometry studies
- Johnson solid demonstrations
- Golden ratio explorations
Art & Design
- Geometric sculptures
- Modern art installations
- Decorative objects
- Jewelry design
Pentagonal Bipyramid Formulas
Volume (V)
Volume with golden ratio √5 for perfect proportions
Surface Area (S)
10 equilateral triangles with √3 relationship
Height (h)
Height with golden ratio for optimal proportions
Johnson Solid
13th Johnson solid in the classification
Pentagonal Bipyramid Parameters
10 equilateral △
7 vertices
15 edges
D5h
All properties are connected to the golden ratio and pentagonal geometry
Calculation Example for a Pentagonal Bipyramid
Given
Find: All properties of the pentagonal bipyramid
1. Volume Calculation
For a = 10:
\[V = \frac{5 + \sqrt{5}}{12} \cdot 10^3\] \[V ≈ \frac{5 + 2.236}{12} \cdot 1000\] \[V ≈ 603.0\]The volume is approximately 603.0 cubic units
2. Surface Area Calculation
For a = 10:
\[S = \frac{5\sqrt{3}}{2} \cdot 10^2\] \[S ≈ 4.33 \cdot 100\] \[S ≈ 433.0\]The surface area is approximately 433.0 square units
3. Height Calculation
For a = 10:
\[h = 2 \sqrt{\frac{5 - \sqrt{5}}{10}} \cdot 10\] \[h ≈ 1.051 \cdot 10\] \[h ≈ 10.51\]The height is approximately 10.51 length units
4. The Perfect Pentagonal Bipyramid
The pentagonal bipyramid with golden ratio beauty
The Pentagonal Bipyramid: Golden Ratio Perfection
The pentagonal bipyramid is a remarkable Johnson solid that embodies the beauty of the golden ratio and pentagonal symmetry. By joining two pentagonal pyramids at their pentagonal bases, this structure creates a unique geometry with 10 congruent equilateral triangles, making it one of the most aesthetically pleasing examples of geometric perfection. The mathematical beauty lies in the elegant relationships involving the golden ratio √5, which connects all geometric properties in harmonious proportions.
The Geometry of the Golden Ratio
The pentagonal bipyramid showcases the perfection of pentagonal geometry:
- Deltahedron: All 10 faces are congruent equilateral triangles
- D5h symmetry: Five-fold rotational symmetry with mirror plane
- Uniformity: All 15 edges have the same length
- Johnson solid: J13 in the classical classification
- Golden ratio: Volume formula contains (5 + √5)
- Pentagonal base: Related to regular pentagon geometry
- Convexity: All vertices point outward
- Aesthetic appeal: Pleasing proportions based on natural ratios
Mathematical Elegance
Golden Ratio Perfection
The formulas of the pentagonal bipyramid are masterpieces of mathematical beauty, with √5 and the golden ratio appearing naturally in the volume calculation, connecting this solid to the fundamental proportions found in nature.
Pentagonal Harmony
The five-fold symmetry creates a perfect balance between complexity and elegance, making this one of the most visually appealing Johnson solids.
Structural Excellence
The combination of pentagonal bases with triangular sides creates optimal stress distribution, making it valuable for architectural and engineering applications.
Natural Inspiration
The pentagonal symmetry appears throughout nature, from flowers to molecular structures, making this solid a bridge between mathematics and the natural world.
Summary
The pentagonal bipyramid represents the perfect marriage of mathematical sophistication and natural beauty. Its structure of ten equilateral triangles, governed by elegant golden ratio relationships, makes it a fascinating subject for mathematicians, architects, and designers. As Johnson solid J13, it demonstrates how pentagonal geometry creates some of the most aesthetically pleasing forms in three-dimensional space. From pure mathematics to practical applications, the pentagonal bipyramid remains a compelling example of how geometric perfection emerges from the fundamental constants of nature, particularly the golden ratio that governs so much of the beauty we see in both mathematics and the natural world.
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