Elongated Square Bipyramid Calculator
Calculator and formulas for an elongated square bipyramid (Johnson solid J15)
Elongated Square Bipyramid Calculator
The Elongated Square Bipyramid
The elongated square bipyramid is a Johnson solid (J15) with 8 triangular faces and 4 square faces.
Johnson Solid J15 Properties
The elongated bipyramid: Two square pyramids joined by a square prism
Elongated Square Bipyramid Structure
The elongated bipyramid with square symmetry.
Johnson solid J15.
What is an elongated square bipyramid?
The elongated square bipyramid is a fascinating Johnson solid:
- Definition: Two square pyramids joined by a square prism
- Johnson solid: J15 in the classification
- Triangular faces: 8 congruent equilateral triangles
- Square faces: 4 congruent squares
- Vertices: 10 vertices total
- Edges: 18 edges (all equal length)
Geometric Properties of the Elongated Square Bipyramid
The elongated square bipyramid shows remarkable geometric properties:
Basic Parameters
- Triangular faces: 8 equilateral triangles
- Square faces: 4 congruent squares
- Vertices: 10 vertices (6 at base, 4 at tips)
- Edges: 18 edges (all equal length)
Special Properties
- Elongation: Extended along one axis
- Square symmetry: 4-fold rotational symmetry
- Uniform edges: All edges equal length
- Convex: No concave edges or vertices
Mathematical Relationships
The elongated square bipyramid follows elegant mathematical laws:
Volume Formula
Combination of square prism and pyramid volumes. Contains √2 relationships.
Surface Area Formula
4 squares plus 8 triangles. Simple additive formula.
Applications of the Elongated Square Bipyramid
Elongated square bipyramids find applications in various fields:
Architecture & Construction
- Tower and spire designs
- Decorative roof elements
- Structural supports
- Architectural details
Science & Technology
- Crystal structures
- Molecular geometry
- Optical components
- Engineering applications
Education & Teaching
- Geometry instruction
- 3D visualization studies
- Johnson solid demonstrations
- Mathematical modeling
Art & Design
- Sculptural works
- Geometric art pieces
- Decorative objects
- Modern design elements
Elongated Square Bipyramid Formulas
Volume (V)
Volume combining prism and pyramid components
Surface Area (S)
4 squares plus 8 equilateral triangles
Height (h)
Height including elongation and pyramid tips
Johnson Solid
15th Johnson solid in the classification
Elongated Square Bipyramid Parameters
8 equilateral △
4 squares □
10 vertices
18 edges
All properties follow from the combination of square prism and pyramid elements
Calculation Example for an Elongated Square Bipyramid
Given
Find: All properties of the elongated square bipyramid
1. Volume Calculation
For a = 10:
\[V = \left(1+\frac{\sqrt{2}}{3}\right) \cdot 10^3\] \[V ≈ 1.4714 \cdot 1000\] \[V ≈ 1471.4\]The volume is approximately 1471.4 cubic units
2. Surface Area Calculation
For a = 10:
\[S = (4 + 2\sqrt{3}) \cdot 10^2\] \[S ≈ 7.4641 \cdot 100\] \[S ≈ 746.41\]The surface area is approximately 746.41 square units
3. Height Calculation
For a = 10:
\[h = \left(\frac{2}{\sqrt{2}} + 1\right) \cdot 10\] \[h ≈ 2.4142 \cdot 10\] \[h ≈ 24.142\]The height is approximately 24.142 length units
4. The Elongated Structure
The elongated square bipyramid with perfect square symmetry
The Elongated Square Bipyramid: Extended Geometric Beauty
The elongated square bipyramid is a remarkable Johnson solid that demonstrates the beauty of geometric extension. By combining two square pyramids with a square prism, this structure creates a unique geometry with both triangular and square faces. The mathematical elegance lies in the harmonious integration of different geometric elements, where the square symmetry governs all proportions. As Johnson solid J15, it showcases how simple geometric transformations can lead to complex and beautiful polyhedra with practical applications.
The Geometry of Extension
The elongated square bipyramid showcases the perfection of geometric elongation:
- Composite structure: Two square pyramids joined by a square prism
- Square symmetry: 4-fold rotational symmetry around central axis
- Mixed faces: 8 equilateral triangles and 4 squares
- Johnson solid: J15 in the classical classification
- Uniform edges: All edges have the same length
- Convex hull: All vertices point outward
- Structural stability: Excellent load distribution properties
Mathematical Elegance
Square Root Relationships
The formulas elegantly combine linear dimensions with √2 and √3 factors, reflecting the underlying square and triangular geometries.
Modular Construction
The elongated structure demonstrates how basic geometric shapes can be combined to create more complex and interesting forms.
Practical Proportions
The elongation creates optimal proportions for many practical applications while maintaining geometric perfection.
Aesthetic Balance
The combination of squares and triangles creates a visually pleasing balance between regularity and complexity.
Summary
The elongated square bipyramid represents the perfect fusion of geometric simplicity and structural sophistication. Its unique combination of square and triangular faces, governed by elegant mathematical relationships, makes it a fascinating subject for mathematicians, architects, and designers. As Johnson solid J15, it demonstrates how geometric extension can transform simple forms into complex, beautiful, and practical structures. From theoretical mathematics to real-world applications, the elongated square bipyramid continues to inspire with its perfect balance of form and function.
|
|