Polyhedron as Polyhedra Generator (segments trisection)

Roman Chijner, Apr 19, 2021

Let the vertices of the initial polyhedron belong to the same sphere. On its basis, can be constructed a certain series of polyhedra. The vertices of each of them are the points of the trisections of the segments of the original polyhedron that have the same length (calculated with a certain accuracy). Obviously, the number of vertices of the constructed polyhedron is twice the number of trisected segments and they all lie on the same sphere.

Table of Contents

  1. ⓿. Biscribed Pentakis Dodecahedron
    1. Biscribed Pentakis Dodecahedron: The Icosahedron-Dodecahedron Compound whose all vertices lie on the same sphere
    2. Series of polyhedra obtained by trisection (truncation) segments of the original polyhedron
    3. Serie of polyhedra obtained by trisection (truncation) segments of the Biscribed Pentakis Dodecahedron
    4. Images. Series of polyhedra obtained by trisection (truncation) different segments of the original polyhedron
  2. ❶. Rhombicosidodecahedron (V=120) and its dual polyhedron- (V=122) from Biscribed Pentakis Dodecahedron for the case of trisection of its 1st-order segments
    1. Rhombicosidodecahedron from Biscribed Pentakis Dodecahedron for the case of trisection of its 1st-order segments
    2. Images . Rhombicosidodecahedron from Biscribed Pentakis Dodecahedron for the case of trisection of its 1st-order segments
    3. Images 1. Rhombicosidodecahedron from Biscribed Pentakis Dodecahedron for the case of trisection of its 1st-order segments
    4. Images 2. Rhombicosidodecahedron from Biscribed Pentakis Dodecahedron for the case of trisection of its 1st-order segments
  3. ❷. Truncated dodecahedron (V=60) and its dual polyhedron- Triakis icosahedron (V=32) from Biscribed Pentakis Dodecahedron for the case of trisection of its 2nd-order segments
    1. Truncated Dodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 2nd-order segments
    2. Images . Truncated dodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 2nd-order segments
    3. Images 1. Truncated dodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 2nd-order segments
    4. Images 2. Truncated dodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 2nd-order segments
  4. ❸. Truncated icosahedron (V=60) and its dual polyhedron- Pentakis dodecahedron (V=32) from Biscribed Pentakis Dodecahedron for the case of trisection of its 3rd-order segments
    1. Truncated Icosahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 3rd-order segments
    2. Images . Truncated icosahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 3rd-order segments
    3. Images 1. Truncated icosahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 3rd-order segments
    4. Images 2. Truncated icosahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 3rd-order segments
  5. ❹. Truncated icosidodecahedron (V=120) and its dual polyhedron- Disdyakis triacontahedron (V=62) from Biscribed Pentakis Dodecahedron for the case of trisection of its 4th-order segments
    1. Truncated icosidodecahedron (V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 4th-order segments
    2. Images . Truncated icosidodecahedron (V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 4th-order segments
    3. Images 1. Truncated icosidodecahedron (V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 4th-order segments
    4. Images 2. Truncated icosidodecahedron (V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 4th-order segments
  6. ❺. Polyhedron(V=120) and its dual polyhedron (V=152) from Biscribed Pentakis Dodecahedron for the case of trisection of its 5th-order segments
    1. Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 5th-order segments
    2. Images . Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 5th-order segments
    3. Images 1. Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 5th-order segments
    4. Images 2. Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 5th-order segments
  7. ❻. Polyhedron(V=120) and its dual polyhedron (V=152) from Biscribed Pentakis Dodecahedron for the case of trisection of its 6th-order segments
    1. Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 6th-order segments
    2. Images . Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 6th-order segments
    3. Images 1. Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 6th-order segments
    4. Images 2. Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 6th-order segments
  8. ❼. The Great Rhombicosidodecahedron (V=120) and its dual Disdyakis triacontahedron (V=62) from Biscribed Pentakis Dodecahedron for the case of trisection of its 7th-order segments
    1. The Great Rhombicosidodecahedron (V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 7th-order segments
    2. Images . The Great Rhombicosidodecahedron (V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 7th-order segments
    3. Images 1. The Great Rhombicosidodecahedron (V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 7th-order segments
    4. Images 2. The Great Rhombicosidodecahedron (V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 7th-order segments
  9. ❽. Rhombicosidodecahedron (V=60) and its dual Deltoidal Hexecontahedron (V=62) from Biscribed Pentakis Dodecahedron for the case of trisection of its 8th-order segments (Variant1)
    1. Rhombicosidodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 8th-order segments(Variant1)
    2. Images . Rhombicosidodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 8th-order segments(Variant1)
    3. Images 1. Rhombicosidodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 8th-order segments(Variant1)
    4. Images 2. Rhombicosidodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 8th-order segments(Variant1)
  10. ❾. Rhombicosidodecahedron (V=60) and its dual Deltoidal Hexecontahedron (V=62) from Biscribed Pentakis Dodecahedron for the case of trisection of its 9th-order segments(Variant2)
    1. Rhombicosidodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 9th-order segments(Variant2)
    2. Images . Rhombicosidodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 9th-order segments(Variant2)
    3. Images 1. Rhombicosidodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 9th-order segments(Variant2)
  11. ❿. Polyhedron(V=120) and its dual Polyhedron(V=152) from Biscribed Pentakis Dodecahedron for the case of trisection of its 10th-order segments
    1. Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of a trisection of its 10th-order segments
    2. Images . Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of a trisection of its 10th-order segments
    3. Images 1. Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of a trisection of its 10th-order segments
    4. Images 2. Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of a trisection of its 10th-order segments
  12. ⓫. Biscribed Pentakis Dodecahedron (V=32) and its dual Biscribed Truncated Icosahedron(V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 11th-order segments
    1. Biscribed Pentakis Dodecahedron (V=32) from Biscribed Pentakis Dodecahedron for the case of a trisection of its 11th-order segments

1.

⓿. Biscribed Pentakis Dodecahedron

  • 1. Biscribed Pentakis Dodecahedron: The Icosahedron-Dodecahedron Compound whose all vertices lie on the same sphere
  • 2. Series of polyhedra obtained by trisection (truncation) segments of the original polyhedron
  • 3. Serie of polyhedra obtained by trisection (truncation) segments of the Biscribed Pentakis Dodecahedron
  • 4. Images. Series of polyhedra obtained by trisection (truncation) different segments of the original polyhedron

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2.

❶. Rhombicosidodecahedron (V=120) and its dual polyhedron- (V=122) from Biscribed Pentakis Dodecahedron for the case of trisection of its 1st-order segments

  • 1. Rhombicosidodecahedron from Biscribed Pentakis Dodecahedron for the case of trisection of its 1st-order segments
  • 2. Images . Rhombicosidodecahedron from Biscribed Pentakis Dodecahedron for the case of trisection of its 1st-order segments
  • 3. Images 1. Rhombicosidodecahedron from Biscribed Pentakis Dodecahedron for the case of trisection of its 1st-order segments
  • 4. Images 2. Rhombicosidodecahedron from Biscribed Pentakis Dodecahedron for the case of trisection of its 1st-order segments

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3.

❷. Truncated dodecahedron (V=60) and its dual polyhedron- Triakis icosahedron (V=32) from Biscribed Pentakis Dodecahedron for the case of trisection of its 2nd-order segments

  • 1. Truncated Dodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 2nd-order segments
  • 2. Images . Truncated dodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 2nd-order segments
  • 3. Images 1. Truncated dodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 2nd-order segments
  • 4. Images 2. Truncated dodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 2nd-order segments

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4.

❸. Truncated icosahedron (V=60) and its dual polyhedron- Pentakis dodecahedron (V=32) from Biscribed Pentakis Dodecahedron for the case of trisection of its 3rd-order segments

  • 1. Truncated Icosahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 3rd-order segments
  • 2. Images . Truncated icosahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 3rd-order segments
  • 3. Images 1. Truncated icosahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 3rd-order segments
  • 4. Images 2. Truncated icosahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 3rd-order segments

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5.

❹. Truncated icosidodecahedron (V=120) and its dual polyhedron- Disdyakis triacontahedron (V=62) from Biscribed Pentakis Dodecahedron for the case of trisection of its 4th-order segments

  • 1. Truncated icosidodecahedron (V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 4th-order segments
  • 2. Images . Truncated icosidodecahedron (V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 4th-order segments
  • 3. Images 1. Truncated icosidodecahedron (V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 4th-order segments
  • 4. Images 2. Truncated icosidodecahedron (V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 4th-order segments

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6.

❺. Polyhedron(V=120) and its dual polyhedron (V=152) from Biscribed Pentakis Dodecahedron for the case of trisection of its 5th-order segments

  • 1. Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 5th-order segments
  • 2. Images . Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 5th-order segments
  • 3. Images 1. Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 5th-order segments
  • 4. Images 2. Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 5th-order segments

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7.

❻. Polyhedron(V=120) and its dual polyhedron (V=152) from Biscribed Pentakis Dodecahedron for the case of trisection of its 6th-order segments

  • 1. Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 6th-order segments
  • 2. Images . Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 6th-order segments
  • 3. Images 1. Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 6th-order segments
  • 4. Images 2. Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 6th-order segments

7.1.

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8.

❼. The Great Rhombicosidodecahedron (V=120) and its dual Disdyakis triacontahedron (V=62) from Biscribed Pentakis Dodecahedron for the case of trisection of its 7th-order segments

  • 1. The Great Rhombicosidodecahedron (V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 7th-order segments
  • 2. Images . The Great Rhombicosidodecahedron (V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 7th-order segments
  • 3. Images 1. The Great Rhombicosidodecahedron (V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 7th-order segments
  • 4. Images 2. The Great Rhombicosidodecahedron (V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 7th-order segments

8.1.

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9.

❽. Rhombicosidodecahedron (V=60) and its dual Deltoidal Hexecontahedron (V=62) from Biscribed Pentakis Dodecahedron for the case of trisection of its 8th-order segments (Variant1)

  • 1. Rhombicosidodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 8th-order segments(Variant1)
  • 2. Images . Rhombicosidodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 8th-order segments(Variant1)
  • 3. Images 1. Rhombicosidodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 8th-order segments(Variant1)
  • 4. Images 2. Rhombicosidodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 8th-order segments(Variant1)

9.1.

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10.

❾. Rhombicosidodecahedron (V=60) and its dual Deltoidal Hexecontahedron (V=62) from Biscribed Pentakis Dodecahedron for the case of trisection of its 9th-order segments(Variant2)

  • 1. Rhombicosidodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 9th-order segments(Variant2)
  • 2. Images . Rhombicosidodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 9th-order segments(Variant2)
  • 3. Images 1. Rhombicosidodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 9th-order segments(Variant2)

10.1.

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11.

❿. Polyhedron(V=120) and its dual Polyhedron(V=152) from Biscribed Pentakis Dodecahedron for the case of trisection of its 10th-order segments

  • 1. Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of a trisection of its 10th-order segments
  • 2. Images . Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of a trisection of its 10th-order segments
  • 3. Images 1. Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of a trisection of its 10th-order segments
  • 4. Images 2. Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of a trisection of its 10th-order segments

11.1.

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12.

⓫. Biscribed Pentakis Dodecahedron (V=32) and its dual Biscribed Truncated Icosahedron(V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 11th-order segments

  • 1. Biscribed Pentakis Dodecahedron (V=32) from Biscribed Pentakis Dodecahedron for the case of a trisection of its 11th-order segments

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