A STEM Project: Constructing Da Vinci's Divine Proportion Polyhedra Models - Part 1 of 2

Plate I - Plate XVIII of Da Vinci's Polyhedra Models

Plate XIX - Plate XXX of Da Vinci's Polyhedra Models

An example of one of Da Vinci's drawings.

Leonardo Da Vinci was a student of Luca Pacioli, an Italian mathematician. As a student, Da Vinci illustrated a book for his teacher called “De Divina Proportione” which means "On the Divine Proportion" in 1498. The book can be viewed online at the Open Library Organization website.

https://openlibrary.org/works/OL16870546W/Divina_proportione?edition=

In this book, Pacioli writes about the mathematics of proportions in polyhedra and the golden ratio. The golden ratio is a value equal to about 1.618. It is based on a line which is divided into two segments and the ratios of the segments are calculated.  The result produces the golden ratio value; it is also known as Phi.  Given a line with a length of 1, divide the line at a point as shown below:

 Phi, 1.618, is a number which appears frequently in nature and our eye is used to seeing this value. The dodecahedron and icosahedron have golden ratios in their dimensions.  A fun movie to watch explaining golden ratios is the Disney movie, Donald in Mathmagic Land (1959).  Here is a link to the movie on Youtube. https://www.youtube.com/watch?v=U_ZHsk0-eF0&t=418s The explanation starts at 7:00.

Leonardo Da Vinci drew fifty nine polyhedra models.  The first thirty will be included in this blog post and the remaining twenty nine polyhedra models will be in the next blog posting which is https://papercraftetc.blogspot.com/2020/08/a-stem-project-constructing-da-vincis_21.html

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

https://drive.google.com/file/d/1WdTvoh7Qzl-8XN7hdraCE69f3Zty_PL2/view?usp=sharing

Here is the .Studio file.

https://drive.google.com/file/d/1nvFtY2wNlT3ajVvIblL1RqDbXN2dn7PT/view?usp=sharing

Here is the SVG.

https://drive.google.com/file/d/1up1XgY_DQ1qvG3D7_YgeWYeUCEFj5_LB/view?usp=sharing

I duplicated Da Vinci's drawings by constructing three-dimensional paper models. In my description of each polyhedron, 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 I, II - Solid & Hollow Plane Tetrahedron	Side Length of 3 inches
Faces	4 equilateral triangles
Edges	6
Vertices	4

Plate I - Solid Plane Tetrahedron

Plate II - Hollow Plane Tetrahedron

Plate III, IV - Solid & Hollow Truncated Tetrahedron	Side Length of 1 inch
Faces	4 equilateral triangles, 4 hexagons
Edges	18
Vertices	12

Plate III - Solid Truncated Tetrahedron

Plate IV - Hollow Truncated Tetrahedron

Plate V, VI - Solid & Hollow Elevated Tetrahedron	Side Length of 1.618 inches
Faces	12 equilateral triangles
Edges	18
Vertices	8

Plate V - Solid Elevated Tetrahedron

Plate VI - Hollow Elevated Tetrahedron

Plate VII, VIII - Solid & Hollow Plane Hexahedron or Cube	Side Length of 1.618 inches
Faces	6 squares
Edges	12
Vertices	8

Plate VII - Solid Plane Hexahedron or Cube

Plate VIII - Hollow Plane Hexahedron or Cube

Plate VIIII, X - Solid & Hollow Truncated Cube	Side Length of 1.618 inches
Faces	6 squares, 8 equilateral triangles
Edges	24
Vertices	12

Plate IX - Solid Truncated Cube

Plate X - Hollow Truncated Cube

Plate XI, XII - Solid & Hollow Elevated Cube	Side Length of 1.618 inches
Faces	24 equilateral triangles
Edges	36
Vertices	14

Plate XI - Solid Elevated Cube

Plate XII - Hollow Elevated Cube

Plate XIII, XIV - Solid & Hollow Elevated Truncated Cube	Side Length of 1.618 inches
Faces	6 quadrilateral pyramids and 8 triangular pyramids which form to make 48 equilateral triangle faces
Edges	72
Vertices	26

Plate XIII - Solid Elevated Truncated Cube

Plate XIV - Hollow Elevated Truncated Cube

Plate XV, XVI - Solid & Hollow Plane Octahedron	Side Length of 1.618 inches
Faces	8 equilateral triangles
Edges	12
Vertices	6

Plate XV - Solid Plane Octahedron 

Plate XVI - Hollow Plane Octahedron 

Plate XVII, XVIII - Solid & Hollow Truncated Octahedron	Side Length of 1.618 inches
Faces	8 hexagons, 6 squares
Edges	36
Vertices	24

Plate XVII - Solid Truncated Octahedron 

Plate XVIII - Hollow Truncated Octahedron 

Plate XIX, XX - Solid & Hollow Elevated Octahedron	Side Length of 1.618 inches
Faces	24 equilateral triangles
Edges	36
Vertices	14

Plate XIV - Solid Elevated Octahedron 

Plate XX - Hollow Elevated Octahedron 

Plate XXI, XXII - Solid & Hollow Plane Icosahedron	Side Length of 1.618 inches
Faces	20 equilateral triangles
Edges	30
Vertices	12

Plate XXI - Solid Plane Icosahedron

Plate XXII - Hollow Plane Icosahedron

Plate XXIII, XXIV - Solid & Hollow Truncated Icosahedron	Side Length of 1.618 inches
Faces	20 hexagons, 12 pentagons
Edges	90
Vertices	60

Plate XXIII - Solid Truncated Icosahedron

Plate XXIV - Hollow Truncated Icosahedron

Plate XXV, XXVI - Solid & Hollow Elevated Icosahedron - aka Great Stellated Dodecahedron	Side Length of 1.618 inches
Faces	60 equilateral triangles
Edges	90
Vertices	32

Plate XXV - Solid Elevated Icosahedron

Plate XXVI - Hollow Elevated Icosahedron

Plate XXVII, XXVIII - Solid & Hollow Plane Dodecahedron	Side Length of 1.618
Faces	12 pentagons
Edges	30
Vertices	20

Plate XXVII - Solid Plane Dodecahedron 

Plate XXVIII - Hollow Plane Dodecahedron 

Plate XXIX, XXX - Solid & Hollow Truncated Dodecahedron	Side Length of 1.618 inches
Faces	12 pentagons, 20 equilateral triangles
Edges	60
Vertices	30

Plate XXIX - Solid Truncated Dodecahedron 

Plate XXX - Hollow Truncated Dodecahedron