Plate I - Plate XVIII of Da Vinci's Polyhedra Models
Plate XIX - Plate XXX of Da Vinci's Polyhedra Models
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.
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:
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.
Here is the .Studio file.
Here is the SVG.
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
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Side Length of 3 inches
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Faces
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4 equilateral triangles
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Edges
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6
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Vertices
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4
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Plate III, IV - Solid & Hollow Truncated Tetrahedron
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Side Length of 1 inch
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Faces
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4 equilateral triangles, 4 hexagons
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Edges
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18
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Vertices
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12
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Plate III - Solid Truncated Tetrahedron
Plate IV - Hollow Truncated Tetrahedron
Plate V, VI - Solid & Hollow Elevated Tetrahedron
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Side Length of 1.618 inches
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Faces
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12 equilateral triangles
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Edges
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18
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Vertices
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8
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Plate V - Solid Elevated Tetrahedron
Plate VI - Hollow Elevated Tetrahedron
Plate VII, VIII - Solid & Hollow Plane Hexahedron or Cube
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Side Length of 1.618 inches
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Faces
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6 squares
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Edges
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12
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Vertices
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8
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Plate VII - Solid Plane Hexahedron or Cube
Plate VIII - Hollow Plane Hexahedron or Cube
Plate VIIII, X - Solid & Hollow Truncated Cube
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Side Length of 1.618 inches
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Faces
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6 squares, 8 equilateral triangles
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Edges
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24
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Vertices
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12
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Plate IX - Solid Truncated Cube
Plate X - Hollow Truncated Cube
Plate XI, XII - Solid & Hollow Elevated Cube
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Side Length of 1.618 inches
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Faces
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24 equilateral triangles
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Edges
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36
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Vertices
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14
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Plate XI - Solid Elevated Cube
Plate XII - Hollow Elevated Cube
Plate XIII, XIV - Solid & Hollow Elevated Truncated Cube
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Side Length of 1.618 inches
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Faces
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6 quadrilateral pyramids and 8 triangular pyramids which form to make 48 equilateral triangle faces
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Edges
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72
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Vertices
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26
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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
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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
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Side Length of 1.618 inches
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Faces
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8 hexagons, 6 squares
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Edges
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36
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Vertices
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24
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Plate XVII - Solid Truncated Octahedron
Plate XVIII - Hollow Truncated Octahedron
Plate XIX, XX - Solid & Hollow Elevated Octahedron
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Side Length of 1.618 inches
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Faces
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24 equilateral triangles
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Edges
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36
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Vertices
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14
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Plate XIV - Solid Elevated Octahedron
Plate XX - Hollow Elevated Octahedron
Plate XXI, XXII - Solid & Hollow Plane Icosahedron
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Side Length of 1.618 inches
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Faces
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20 equilateral triangles
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Edges
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30
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Vertices
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12
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Plate XXI - Solid Plane Icosahedron
Plate XXII - Hollow Plane Icosahedron
Plate XXIII, XXIV - Solid & Hollow Truncated Icosahedron
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Side Length of 1.618 inches
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Faces
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20 hexagons, 12 pentagons
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Edges
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90
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Vertices
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60
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Plate XXIII - Solid Truncated Icosahedron
Plate XXIV - Hollow Truncated Icosahedron
Plate XXV, XXVI - Solid & Hollow Elevated Icosahedron - aka Great Stellated Dodecahedron
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Side Length of 1.618 inches
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Faces
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60 equilateral triangles
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Edges
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90
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Vertices
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32
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Plate XXV - Solid Elevated Icosahedron
Plate XXVI - Hollow Elevated Icosahedron
Plate XXVII, XXVIII - Solid & Hollow Plane Dodecahedron
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Side Length of 1.618
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Faces
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12 pentagons
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Edges
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30
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Vertices
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20
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Plate XXVII - Solid Plane Dodecahedron
Plate XXVIII - Hollow Plane Dodecahedron
Plate XXIX, XXX - Solid & Hollow Truncated Dodecahedron
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Side Length of 1.618 inches
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Faces
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12 pentagons, 20 equilateral triangles
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Edges
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60
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Vertices
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30
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Plate XXIX - Solid Truncated Dodecahedron
Plate XXX - Hollow Truncated Dodecahedron
Hello,
ReplyDeleteThese are wonderful, thank you for posting. I was wondering if the svg file could be split into two files perhaps, my computer is having difficulty opening such a large file.
Thank you!
I have provided a PDF of the file. You can convert the PDF to an SVG.
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