A STEM Project: More Inscribed Polyhedra Based on Pacioli's Divine Proportion Book

In the style of Da Vinci's polyhedra model drawings from Pacioli's Divine Proportion book, I inscribed more polyhedra in this blog posting.  I think these are excellent models for students to explore the concept of stellating and truncating Platonic solids.

The top row is the original Platonic solid with their truncated solid inside. 

The bottom row is a stellated Platonic solid with their original Platonic solid inside. 

From left to right in both columns, the models are a tetrahedron, cube, octahedron, dodecahedron and icosahedron.

Here is the PDF. I used 65 lb. cardstock to make the models.

https://drive.google.com/file/d/1tqSb1XW-AN_Gg_H2Pmo0uG8NTBFMPv-Z/view?usp=sharing

Here is the .Studio file.

https://drive.google.com/file/d/1acmLW5CihugVom1mGgBKV35jMeaXcBsj/view?usp=sharing

Here is the SVG.

https://drive.google.com/file/d/1JBfYrkbxdg_icao6RQt9ZnniT66sq3SF/view?usp=sharing

A truncated tetrahedron inscribed inside of tetrahedron. 

A tetrahedron inscribed in a hollow elevated tetrahedron.

A truncated cube inscribed in a hollow cube.

A solid cube inscribed in a hollow elevated cube.

A solid truncated octahedron inscribed in a hollow plane octahedron.

A solid plane octahedron inscribed in a hollow elevated octahedron.

A solid truncated dodecahedron inscribed in a hollow plane dodecahedron.

A solid plane dodecahedron inscribed in a hollow elevated dodecahedron.

A solid truncated icosahedron inscribed in a hollow plane icosahedron.

A solid plane icosahedron inscribed in a hollow elevated icosahedron.

A STEM Project: Inscribing Da Vinci's Divine Proportion Polyhedra Models

Platonic solids inscribed into another Platonic solid using maximal configuration.

Computer generated models from Moritz Firsching's article about computing maximal side lengths of inscribed polytopes. *

After making Da Vinci's divine proportion polyhedra models, see blog posts https://papercraftetc.blogspot.com/2020/08/a-stem-project-constructing-da-vincis.html and https://papercraftetc.blogspot.com/2020/08/a-stem-project-constructing-da-vincis_21.html, I decided to inscribe the polyhedra models into one another.  Pacioli wrote in his Divine Proportion book about these inscriptions.

The five Platonic solids, the tetrahedron, cube, octahedron, icosahedron and dodecahedron, can be derived from a single one, the dodecahedron, which, according to Pacioli, "sustains the existence of all the others and governs the manifold harmonies and interrelations among all five". Since the dodecahedron is the basis for all others, Pacioli claimed that it would be only mathematically possible with a specific proportion which he named the "Divine Proportion".

Pacioli wrote about inscriptions of the five Platonic solids solids using the sphere method of calculating the side lengths of the interior polyhedron.  I inscribed the Platonic solids as I thought Da Vinci would have done if he had the Silhouette software and cutting technology. 

 I found out while making the models that as the inscribed polyhedron’s size approached a sphere, the size calculations became more complex and difficult to calculate. With some investigation on the internet, I discovered an article about maximizing the side lengths of the interior polyhedron. The computations for six of the polyhedra were just calculated in 2018 by the article's author, Moritz Firsching.  These maximal size calculations were in the annals of unsolved geometric problems for many centuries. Using Firsching’s calculations, I was able to complete the 20 inscribed Platonic solid models.

"Computing maximal copies of polytopes contained in a polytope", by Moritz Firsching, Institut fu ̈r Mathematik FU Berlin Arnimallee 2 14195 Berlin Germany, July 16, 2018 https://arxiv.org/pdf/1407.0683.pdf *

The following calculations were cited in this paper:

Maximum side lengths of polyhedron inscribed in another polyhedron.

Using the above calculations, I created my models. I used Glue Dots to affix the inscribed polyhedra. I also used clear acetate to make a base for some of the models so that I could adhere the inscribed polyhedra model to something since the point of contact was sometimes in mid-space in the outer hollow polyhedron.

 

Here is the PDF.  I used 65lb. cardstock to make the models.

https://drive.google.com/file/d/1iDsHsBYoMphk455tLWf2wuO57xXTEvOw/view?usp=sharing 

Here is the .Studio file.

https://drive.google.com/file/d/1aABrjbhXo7eHu2nSUxUl9iu4xiAmiCL8/view?usp=sharing

Here is the SVG.

https://drive.google.com/file/d/1YUDG1c--YuN4wJ_xHuC8D2ZFLKo2Vdti/view?usp=sharing

Polyhedron Inscribed in a Tetrahedron  

I made the side length of the Tetrahedron 3 inches in order to get a larger sized model. 

1). Cube in a Tetrahedron. The side length of the cube was calculated as 0.296 x 3 = 0.888 inches.

0.888 in. side length Cube in a 3 in. side length Tetrahedron

2). Octahedron in a Tetrahedron. The side length of the octahedron was calculated as 0.500 x 3 = 1.5 inches. 

1.5 in. side length Octahedron in a 3 in. side length Tetrahedron

3).  Dodecahedron in a Tetrahedron.  The side length of the dodecahedron is calculated as 0.163 x 3 = 0.489 inches. 

0.489 in. side length Dodecahedron in a 3 in. side length Tetrahedron

4).  Icosahedron in a Tetrahedron.  The side length of the icosahedron is calculated as 0.27 x 3 = 0.81 inches. 

0.81 in. side length Icosahedron in a 3 in. side length Tetrahedron

Polyhedron Inscribed in a Cube 

I made the side length of the Cube 2 inches in order to get a larger sized model. 

5).  Tetrahedron in a Cube. The side length of the tetrahedron is calculated as 1.414 x 2 = 2.828 inches.

2.828 in. side length Tetrahedron in a 2 in. side length Cube

6). Octahedron in a Cube. The side length of the octahedron is calculated as 1.06 x 2 = 2.12 inches.

2.12 in. side length Octahedron in a 2 in. side length Cube

7). Dodecahedron in a Cube. The side length of the dodecahedron is calculated as 0.394 x 2 = 0.788 inches.

0.788 in. side length dodecahedron in a 2 in. side length Cube

8). Icosahedron in a Cube. The side length of the icosahedron is calculated as 0.618 x 2 = 1.236 inches.

1.236 in. side length icosahedron in a 2 in. side length Cube

Polyhedron Inscribed in an Octahedron

I made the side length of the Octahedron 3 inches in order to get a larger sized model. 

9).  Tetrahedron in an Octahedron. The side length of the tetrahedron is calculated as 1 x 3 = 3 inches.

3 in. side length Tetrahedron in a 3 in. side length Octahedron 

10). Cube in an Octahedron. The side length of the cube is calculated as 0.586 x 3 = 1.758 inches.

1.758 in. side length cube  in a 3 in. side length Octahedron

11). Dodecahedron in an Octahedron. The side length of the dodecahedron is calculated as 0.313 x 3 = 0.939 inches.

0.939 in. side length dodecahedron in a 3 in. side length Octahedron

12). Icosahedron in an Octahedron. The side length of the icosahedron is calculated as 0.54 x 3 = 1.62 inches.

1.62 in. side length icosahedron in a 3 in. side length Octahedron

Polyhedron Inscribed in a Dodecahedron

The side length of the Dodecahedron is 1 inch

13).  Tetrahedron in a Dodecahedron. 

2.288 in. side length Tetrahedron in a 1 in. side length Dodecahedron

14). Cube in a Dodecahedron. 

1.618 in. side length cube  in a 1 in. side length Dodecahedron

15). Octahedron in a Dodecahedron. 

1.851 in. side length octahedron in a 1 in. side length Dodecahedron

16). Icosahedron in a Dodecahedron. 

1.309 in. side length icosahedron in a 1 in. side length Dodecahedron

Polyhedron Inscribed in an Icosahedron

I made the side length of the Icosahedron 2 inches in order to get a larger sized model. 

17).  Tetrahedron in an Icosahedron. The side length of the tetrahedron is calculated as 1.347 x 2 = 2.694 inches.

 

2.694 in. side length Tetrahedron in a 2 in. side length Icosahedron

18). Cube in an Icosahedron. The side length of the cube is calculated as 0.939 x 2 = 1.878 inches.

1.878 in. side length cube in a 2 in. side length Icosahedron

19). Octahedron in an Icosahedron. The side length of the octahedron is calculated as 1.181 x 2 = 2.362 inches.

2.362 in. side length octahedron in a 2 in. side length Icosahedron

20). Dodecahedron in an Icosahedron. The side length of the dodecahedron is calculated as 0.58 x 2 = 1.16 inches.

1.16 in. side length Dodecahedron in a 2 in. side length Icosahedron

A STEM Project: Constructing Da Vinci's Divine Proportion Polyhedra Models - Part 2 of 2

Plate XXXI - Plate XXXVIII of Da Vinci's Polyhedra Models

Plate XL - Plate LIX of Da Vinci's Polyhedra Models

Leonardo Da Vinci drew fifty nine polyhedra models.  The first thirty were included in my last blog post https://www.blogger.com/blog/post/edit/5237417144570156489/3612955597216858264. The remaining twenty nine polyhedra models will be posted here.

Here is the PDF. I used 65lb. cardstock and vellum.

https://drive.google.com/file/d/1890TkEcAqpjCcOz6iGdXci5MvWYTMJyr/view?usp=sharing

Here is the .Studio file.

https://drive.google.com/file/d/1J63ZmmIp0L1IzCzyp53lUVwX3q2f-6JY/view?usp=sharing

Here is the SVG.

https://drive.google.com/file/d/1LCOpcfZrz63rGjsE4ulHXwRKXC7Oh8gk/view?usp=sharing

I duplicated Da Vinci's drawings by constructing three-dimensional paper models. In my description, I will give the side length measure and describe each model based on the number of faces, edges and vertices using Euler's formula.  For any convex polyhedron, the number of vertices and faces together is exactly two more than the number of edges.   Vertices + Faces - Edges = 2

A face is a flat, two-dimensional surface that serves as one side of a polyhedron.

An edge is a line segment where two faces meet.

Vertices is the plural of vertex.  Vertices are corner points which are formed by the intersection of faces.

Plate XXXI, XXXII - Solid & Hollow Elevated Dodecahedron aka Small Stellated Dodecahedron	Side Length of 1.618 inches
Faces	60 equilateral triangles
Edges	90
Vertices	32

Plate XXXI - Solid Elevated Dodecahedron

Plate XXXII -Hollow Elevated Dodecahedron

Plate XXXIII, XXXIV - Solid & Hollow Elevated  Truncated Dodecahedron	Side Length of 1.618 inches
Faces	120 equilateral triangles
Edges	180
Vertices	62

Plate XXXIII - Solid Elevated Truncated Dodecahedron

Plate XXXIV - Hollow Elevated Truncated Dodecahedron

Plate XXXV, XXXVI - Solid & Hollow Icosahexahedron aka Rhombicuboctahedron or Small Rhombic Cuboctahedron	Side Length of 1.618 inches
Faces	18 squares, 8 equilateral triangles
Edges	48
Vertices	24

Plate XXXV - Solid Icosahexahedron

Plate XXXVI - Hollow Icosahexahedron

Plate XXXVII, XXXVIII - Solid & Hollow Elevated Icosahexahedron aka Augmented Rhombicuboctahedron or Moravian Star	Side Length of 1.618 inches
Faces	96 equilateral triangles
Edges	144
Vertices	50

Plate XXXVII - Solid Elevated Icosahexahedron

Plate XXXVIII - Hollow Elevated Icosahexahedron

Plate XXXIX, XL - Solid & Hollow Septuagintaduarium Basium	Varying side lengths
Faces	24 isosceles triangles, 48 trapezoids
Edges	132
Vertices	62

Plate XXXIX - Solid Septuagintaduarium Basium

Plate XL - Hollow Septuagintaduarium Basium

Plate XLI, XLII  - Solid & Hollow Triangular Lateral Column	3.725 in. x 1.618 in.
Faces	2 equilateral triangles, 3 rectangles
Edges	9
Vertices	6

Plate XLI - Solid Lateral Triangular Column

Plate XLII - Hollow Lateral Triangular Column

Plate XLIII, XLIV  - Solid & Hollow Triangular Lateral Pyramid	3.725 in.height x 1.618 in.base
Faces	1 equilateral triangle, 3 isosceles triangles
Edges	6
Vertices	4

Plate XLIII - Solid Lateral Triangular Pyramid

Plate XLIV - Hollow Lateral Triangular Pyramid

Plate XLV, XLVI  - Solid & Hollow Quadrangular Lateral Column	3 in. x 1.618 in.
Faces	2 squares, 4 rectangles
Edges	12
Vertices	8

Plate XLV - Solid Quadrangular Lateral Column

Plate XLVI - Hollow Quadrangular Lateral Column

Plate XLVII, XLVIII  - Solid & Hollow Quadrangular Lateral Pyramid	3 in. X 1.618 in.
Faces	1 square, 4 isosceles triangles
Edges	8
Vertices	5

Plate XLVII - Solid Quadrangular Lateral Pyramid

Plate XLVIII - Hollow Quadrangular Lateral Pyramid

Plate XLIX, L - Solid & Hollow Pentagonal Lateral Column	4.3 in. x 1.618 in.
Faces	2 pentagons, 5 rectangles
Edges	15
Vertices	10

Plate XLIX - Solid Pentagonal Lateral Column

Plate L - Hollow Pentagonal Lateral Column

Plate LI, LIII- Solid & Hollow Pentagonal Lateral Pyramid	4.3 in. x 1.618 in.
Faces	1 pentagon, 5 isosceles triangles
Edges	10
Vertices	6

Plate LI - Solid Pentagonal Lateral Pyramid

Plate LII -Hollow Pentagonal Lateral Pyramid

Plate LIII, LIV - Solid & Hollow Hexagonal Lateral Column	5.2 in. x 1.618 in.
Faces	2 hexagons, 6 rectangles
Edges	18
Vertices	12

Plate LIII - Solid Hexagonal Lateral Column

Plate LIV - Hollow Hexagonal Lateral Column

Plate LIV, LV - Solid & Hollow Inequilateral Triangular Lateral Pyramid	Varying sizes
Faces	4 triangles
Edges	6
Vertices	4

Plate LV - Solid Inequilateral Triangular Lateral Pyramid

Plate LVI - Hollow Inequilateral Triangular Lateral Pyramid

Plate LVII - Solid Round Column	4.5 in. x 1.618 in.
Faces	2 circles
Edges	0
Vertices	0

Plate LVII - Solid Round Column

Plate LVIII - Solid Cone	4.5 in. x 1.618 in.
Faces	1 circle
Edges	0
Vertices	1

Plate LVIII - Solid Cone

Plate LVII - Solid Sphere	4.5 inch diameter
Faces	0
Edges	0
Vertices	0

Plate LIX - Solid Sphere - This sphere was made out of vellum paper because vellum is thinner and more flexible than cardstock.