Antiprism: Definitions and Examples

Antiprism: Definitions, Formulas, & Examples

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    Antiprisms are three-dimensional geometric shapes with two congruent polygonal bases that are connected by perpendicular edges. They are known for their ability to reflect light in a way that creates a rainbow of colors, making them popular in decorative applications such as jewelry and art installations. In this article, we will define anti prisms, provide five examples of their use, and include a quiz to test your understanding of these fascinating objects.

    Definitions:

    • Anti prism: A three-dimensional geometric shape with two congruent polygonal bases that are connected by perpendicular edges.
    • Polygonal base: A flat, two-dimensional shape with straight sides and angles.
    • Congruent: Having the same size and shape.

    Five Examples of Antprisms:

    1. Jewelry: Anti prisms are often used as decorative elements in jewelry, such as pendants and earrings. The reflections of light off the geometric shapes create a mesmerizing effect that draws the eye.
    2. Art installations: Anti prisms can also be used to create striking visual displays in art installations. By suspending multiple anti prisms from the ceiling or arranging them in a specific pattern, artists can create a dynamic and visually striking display that engages the viewer.
    3. Home decor: Anti prisms can also be incorporated into home decor, such as lamp shades or vases. These objects can add a touch of whimsy and interest to any room.
    4. Science demonstrations: Anti prisms are sometimes used in science demonstrations to illustrate the principles of light refraction and reflection. By shining a light through an anti prism, students can see how the light is bent and reflected to create a rainbow of colors.
    5. Games: Anti prisms have also been used as game pieces in some board games and puzzle games. The unique shape and colorful reflections of these objects make them an interesting and challenging element to incorporate into game-play.

    Quiz:

    1. What is an anti prism? a) A three-dimensional shape with two congruent polygonal bases connected by perpendicular edges b) A flat, two-dimensional shape with straight sides and angles c) A solid object with a circular base and a pointed top d) A geometric shape with one polygonal base and curved sides
    2. What is a polygonal base? a) A three-dimensional shape with two congruent polygonal bases connected by perpendicular edges b) A flat, two-dimensional shape with straight sides and angles c) A solid object with a circular base and a pointed top d) A geometric shape with one polygonal base and curved sides
    3. What does it mean for two shapes to be congruent? a) They have the same size and shape b) They have the same size but different shapes c) They have different sizes but the same shape d) They have different sizes and shapes
    4. In which of the following applications might you see an anti prism? a) Jewelry b) Art installations c) Home decor d) All of the above
    5. How are anti prisms sometimes used in science demonstrations? a) To illustrate the principles of light refraction and reflection b) To demonstrate the concept of gravity c) To model the movement of planets in our solar system d) To show how electricity flows through a circuit
    6. Can anti prisms be used as game pieces in board games or puzzle games? a) Yes b) No
    7. What is the defining characteristic of an anti prism? a) Its congruent

    Antiprism:

    Polyhedra with available data

    equilateral square antiprism | equilateral pentagonal antiprism | equilateral hexagonal antiprism | equilateral heptagonal antiprism | equilateral octagonal antiprism | equilateral nonagonal antiprism | equilateral decagonal antiprism | regular octahedron (total: 8)

    Visual representations

    Visual representations

    Combinatorial properties

     | vertices | edges | faces
equilateral square antiprism | 8 | 16 | 10 (8 triangles, 2 quadrilaterals)
equilateral pentagonal antiprism | 10 | 20 | 12 (10 triangles, 2 pentagons)
equilateral hexagonal antiprism | 12 | 24 | 14 (12 triangles, 2 hexagons)
equilateral heptagonal antiprism | 14 | 28 | 16 (14 triangles, 2 heptagons)
equilateral octagonal antiprism | 16 | 32 | 18 (16 triangles, 2 octagons)
equilateral nonagonal antiprism | 18 | 36 | 20 (18 triangles, 2 nonagons)
equilateral decagonal antiprism | 20 | 40 | 22 (20 triangles, 2 decagons)
regular octahedron | 6 | 12 | 8 (8 triangles)

    Edge lengths

    1 (16 edges)

    1 (20 edges)

    1 (24 edges)

    1 (28 edges)

    1 (32 edges)

    1 (36 edges)

    1 (40 edges)

    1 (12 edges)

    Geometric properties

     | volume
equilateral square antiprism | ((2 - 1/(1 + 1/sqrt(2)))^(3/2) cot(π/8))/(3 sqrt(2))
equilateral pentagonal antiprism | 5/12 sqrt(1/2 (5 + 2 sqrt(5))) (2 - 1/(1 + 1/4 (1 + sqrt(5))))^(3/2)
equilateral hexagonal antiprism | ((2 + sqrt(3)) (2 - 1/(1 + sqrt(3)/2))^(3/2))/(2 sqrt(2))
equilateral heptagonal antiprism | (7 (2 - 1/(1 + cos(π/7)))^(3/2) cot(π/14))/(12 sqrt(2))
equilateral octagonal antiprism | 1/3 sqrt(2) (2 - 1/(1 + cos(π/8)))^(3/2) cot(π/16)
equilateral nonagonal antiprism | (3 (2 - 1/(1 + cos(π/9)))^(3/2) cot(π/18))/(4 sqrt(2))
equilateral decagonal antiprism | (5 (2 - 1/(1 + sqrt(5/8 + sqrt(5)/8)))^(3/2) cot(π/20))/(6 sqrt(2))
regular octahedron | sqrt(2)/3
 | surface area
equilateral square antiprism | 2 (1 + sqrt(3))
equilateral pentagonal antiprism | 5/2 (sqrt(3) + sqrt(1 + 2/sqrt(5)))
equilateral hexagonal antiprism | 6 sqrt(3)
equilateral heptagonal antiprism | 7/2 (sqrt(3) + cot(π/7))
equilateral octagonal antiprism | 4 (sqrt(3) + cot(π/8))
equilateral nonagonal antiprism | 9/2 (sqrt(3) + cot(π/9))
equilateral decagonal antiprism | 5 (sqrt(3) + sqrt(5 + 2 sqrt(5)))
regular octahedron | 2 sqrt(3)
 | circumradius
equilateral square antiprism | 1/4 sqrt(4 + csc^2(π/8))
equilateral pentagonal antiprism | 1/4 sqrt(4 + (1 + sqrt(5))^2)
equilateral hexagonal antiprism | 1/4 sqrt(4 + 2 (1 + sqrt(3))^2)
equilateral heptagonal antiprism | 1/4 sqrt(4 + csc^2(π/14))
equilateral octagonal antiprism | 1/4 sqrt(4 + csc^2(π/16))
equilateral nonagonal antiprism | 1/4 sqrt(4 + csc^2(π/18))
equilateral decagonal antiprism | 1/4 sqrt(4 + csc^2(π/20))
regular octahedron | 1/sqrt(2)
 | midradius
equilateral square antiprism | 1/4 csc(π/8)
equilateral pentagonal antiprism | 1/4 (1 + sqrt(5))
equilateral hexagonal antiprism | (1 + sqrt(3))/(2 sqrt(2))
equilateral heptagonal antiprism | 1/4 csc(π/14)
equilateral octagonal antiprism | 1/4 csc(π/16)
equilateral nonagonal antiprism | 1/4 csc(π/18)
equilateral decagonal antiprism | 1/4 csc(π/20)
regular octahedron | 1/2
 | inradius
regular octahedron | 1/sqrt(6)
(assuming unit edge lengths)

    Nets

    Nets

    Skeleton graphs

     | skeleton graph name
equilateral square antiprism | 4-antiprism graph
equilateral pentagonal antiprism | 5-antiprism graph
equilateral hexagonal antiprism | 6-antiprism graph
equilateral heptagonal antiprism | 7-antiprism graph
equilateral octagonal antiprism | 8-antiprism graph
equilateral nonagonal antiprism | 9-antiprism graph
equilateral decagonal antiprism | 10-antiprism graph
regular octahedron | octahedral graph

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