Platic Solids and the Five-Intersecting Tetrahedra: 3D Geometry Modeling, Study Guides, Projects, Research of School management&administration

Download Platic Solids and the Five-Intersecting Tetrahedra: 3D Geometry Modeling and more Study Guides, Projects, Research School management&administration in PDF only on Docsity! Platonic Solids and the Five-Intersecting Tetrahedra Page 1 of 21 Stephenie Swope When I first saw the 5-intersecting tetrahedron through my Honors class website, I found it to be beautiful and interesting. As I searched for project ideas, I kept coming back to it and knew I had to explore it further… THEORY AND CALCULATIONS I began with the five Platonic Solids: the cube, octahedron, icosohedron, dodecahedron, and tetrahedron. The properties that determine each platonic solid are that these convex polyhedra are composed of regular polygons of the same type and all of the corners on each solid are all the same. The Platonic Solids are also called the regular solids or regular polyhedra. Another important feature of the Platonic Solids is duality or reciprocity. The principle of duality states that for every polyhedron there exists another polyhedron in which faces and vertices are in complementary locations. This duality pairs the Platonic Solids with one another. In order to determine the dual of a polyhedron, mark a point in the center of each face. Connect each point with lines, which will form the edges of the dual polyhedron. The reverse is also true. This reciprocity defines the duality of the two polyhedra. For example, by marking the center of the 6 faces of a cube and connecting the points, and octahedron is formed. Forming a Cube centered at the origin of three-dimensional space (0,0,0) with a distance of one unit to each point, the vertices are: (-1,-1,-1) (-1,-1,1) (-1,1,-1) (-1,1,1) (1,-1,-1) Platonic Solids and the Five-Intersecting Tetrahedra Page 2 of 21 Stephenie Swope (1,-1,1) (1,1,-1) (1,1,1) Therefore, the centers of each face would have the following centers, which are the vertices of the dual octahedron: (0,0,-1) (0,0,1) (-1,0,0) (1,0,0) (0,-1,0) (0,1,0) Since the cube and the octahedron are duals, taking the centers of the faces of the octahedron would give us the another (smaller) cube, we must use another method to develop another Platonic Solid. This can be done by expanding the points into equal line segments such that the faces of the octahedron become equilateral triangle faces. The length of this edge can be determined knowing this information. For example, the coordinates of the points I and J will be I = (t, 0,−1) and J = (−t, 0,−1), where t is a positive number between 0 and 1. Platonic Solids and the Five-Intersecting Tetrahedra Page 5 of 21 Stephenie Swope Using this data and substituting 2 15  t gives us approximations of the points of all of the vertices of the dodecahedron. Knowing these vertices, it was necessary to determine which points were collinear. This can be done using the distances between the vertices. The edges will be the shortest distances (0.7). Diagonals across the face will be the second distance (1.1). The third distance (1.5) will be the inside distance between the points that can be found by following one edge and one diagonal, or one diagonal then one edge. The fourth distance (1.9) is the Platonic Solids and the Five-Intersecting Tetrahedra Page 6 of 21 Stephenie Swope inside distance between the points that can be found by traveling across one diagonal then another diagonal. I plotted the distances on a table and highlighted the shortest distance to find out which points shared an edge. It is expected that each point will connect to 3 others, and those connections are shown in the table above. The relationships are shown in the table below: Platonic Solids and the Five-Intersecting Tetrahedra Page 7 of 21 Stephenie Swope Once the edges were determined, the points that made up each face of the dodecahedron had to be determined. Using the same method, choose a point. from that point, 2 of the three adjacent edges are chosen. From the point, there are 6 points that are of distance 1.1 (diagonal across a face). These are in pairs. Seeing a diagram of the faces laid out flat will help with this visualization. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 17 18 19 20 2 2 4 1 1 2 4 13 15 18 5 3 12 7 6 11 19 17 16 20 8 9 10 9 14 14 12 13 3 5 18 15 6 11 10 19 17 20 7 16 8 16 Looking again at the distance table, we see that there are indeed, 6 points a distance of 1.1 from each vertex. Platonic Solids and the Five-Intersecting Tetrahedra Page 10 of 21 Stephenie Swope This tetrahedron has 4 vertices and the dodecahedron has 20 vertices. We could inscribe 5 distinct tetrahedra inside a dodecahedron. This is shown here, first without the dodecahedron frame, then with it: This beautiful and symmetrical mathematical object is called the five-intersecting tetrahedral. MODEL DEVELOPMENT The model that I built for the class was developed from Make Shapesi. This model, which will be shown in class, was scanned into the computer. I then drew one section of the model using Printmaster 11 software, copied that image 3 times, and converted the document to a picture. I moved the picture into Micorsoft Word so that I could resize it until I was happy with the size. I separated the corners so that I could print each tetrahedron on different color card stock. There were 4 images to a page because there are four corners in a tetrahedron. I resized the images because I wanted a larger model. Here is a picture of one of the sections: Platonic Solids and the Five-Intersecting Tetrahedra Page 11 of 21 Stephenie Swope POV-Ray Code The POV-Rayii code was developed using the methods and mathematics described above in “Theory and Calculations”. The code used to develop the Cube -> Octahedron -> Icosohedron picturesiii can be found on the HON301 course website. The code used to develop the Dodecahedron, the 5-intersecting tetrahedral, and the movie is below: //Dodecahedron DEFINED BY THE COORDINATES OF ITS VERTICES AND POLYGONS //Files with predefined colors and textures #include "colors.inc" #include "glass.inc" #include "golds.inc" #include "metals.inc" #include "stones.inc" #include "woods.inc" //Place the camera camera { sky <0,0,1> //Don't change this direction <-1,0,0> //Don't change this right <-4/3,0,0> //Don't change this location <30,10,10> //Camera location look_at <0,0,0> //Where camera is pointing angle 5 //Angle of the view--increase to see more, decrease to see less } //Ambient light to "brighten up" darker pictures global_settings { ambient_light White*2 } //Place a light--you can have more than one! light_source {<10,5,10> color White*2 } light_source {<10,-5,10> color White*2 } light_source {<-10,5,10> color White*2 } //Set a background color background { color VLightGrey } //Construct a docecahedron by describing its polygons //The 20 vertices #declare tt=(sqrt(5)-1)/2; #declare w1=<tt,0,-1>; #declare w2=<-tt,0,-1>; #declare w3=<0,-1,tt>; #declare w4=<0,-1,-tt>; #declare w5=<-1,tt,0>; #declare w6=<-1,-tt,0>; Platonic Solids and the Five-Intersecting Tetrahedra Page 12 of 21 Stephenie Swope #declare w7=<0,1,tt>; #declare w8=<0,1,-tt>; #declare w9=<1,tt,0>; #declare w10=<1,-tt,0>; #declare w11=<tt,0,1>; #declare w12=<-tt,0,1>; #declare v1 = (w1+w8+w9)/3; #declare v2 = (w1+w9+w10)/3; #declare v3 = (w1+w4+w10)/3; #declare v4 = (w1+w2+w8)/3; #declare v5 = (w1+w2+w4)/3; #declare v6 = (w7+w11+w12)/3; #declare v7 = (w5+w7+w12)/3; #declare v8 = (w5+w6+w12)/3; #declare v9 = (w3+w6+w12)/3; #declare v10 = (w3+w11+w12)/3; #declare v11 = (w3+w10+w11)/3; #declare v12 = (w9+w10+w11)/3; #declare v13 = (w7+w9+w11)/3; #declare v14 = (w7+w8+w9)/3; #declare v15 = (w5+w7+w8)/3; #declare v16 = (w2+w5+w6)/3; #declare v17 = (w2+w4+w6)/3; #declare v18 = (w2+w5+w8)/3; #declare v19 = (w3+w4+w10)/3; #declare v20 = (w3+w4+w6)/3; //The 12 pentagonal faces #declare fd1 = polygon { 6, v1, v4, v18, v15, v14, v1 texture{pigment {color rgbf <0,0,1,0>}}}; #declare fd2 = polygon { 6, v1, v2, v3, v5, v4, v1 texture{pigment {color rgbf <0,1,0,0>}}}; #declare fd3 = polygon { 6, v1, v2, v12, v13, v14, v1 texture{pigment {color rgbf <1,0,0,0>}}}; #declare fd4 = polygon { 6, v2, v3, v19, v11, v12, v2 texture{pigment {color rgbf <0,0,.5,0>}}}; #declare fd5 = polygon { 6, v3, v5, v17, v20, v19, v3 texture{pigment {color rgbf <0,.5,0,0>}}}; #declare fd6 = polygon { 6, v4, v5, v17, v16, v18, v4 texture{pigment {color rgbf <.5,0,0,0>}}}; #declare fd7 = polygon { 6, v7, v8, v16, v18, v15, v7 texture{pigment {color rgbf <0,.5,1,0>}}}; #declare fd8 = polygon { 6, v6, v10, v11, v12, v13, v6 texture{pigment {color rgbf <.5,0,1,0>}}}; #declare fd9 = polygon { 6, v6, v7, v15, v14, v13, v6 texture{pigment {color rgbf <.5,1,0,0>}}}; #declare fd10 = polygon { 6, v6, v7, v8, v9, v10, v6 texture{pigment {color rgbf <1,0,.5,0>}}}; #declare fd11 = polygon { 6, v8, v9, v20, v17, v16, v8 texture{pigment {color rgbf <1,.5,0,0>}}}; #declare fd12 = polygon { 6, v9, v10, v11, v19, v20, v9 texture{pigment {color rgbf <0,.5,1,0>}}}; //Unite the faces to form the dodecahedron #declare mydodecahedron = object { union { object{fd1} object{fd2} object{fd3} object{fd4} object{fd5} object{fd6} object{fd7} object{fd8} object{fd9} object{fd10} object{fd11} object{fd12} } } //List the defined object(s) to be displayed mydodecahedron //FIVE INTERSECTING TETRAHEDRA DEFINED BY THE COORDINATES OF ITS VERTICES AND POLYGONS //Files with predefined colors and textures #include "colors.inc" #include "glass.inc" #include "golds.inc" #include "metals.inc" #include "stones.inc" #include "woods.inc" //Place the camera camera { sky <0,0,1> //Don't change this Platonic Solids and the Five-Intersecting Tetrahedra Page 15 of 21 Stephenie Swope object{f5_2} object{f5_3} object{f5_4} } pigment { Yellow } }; //Intersect the 5 tetrahedron merge{ object{mytetra1} object{mytetra2} object{mytetra3} object{mytetra4} object{mytetra5} } //FIVE TETRAHEDRA MOVIE //ANIMATE WITH CLOCK FROM 0 TO 7 //Files with predefined colors and textures #include "colors.inc" #include "glass.inc" #include "golds.inc" #include "metals.inc" #include "stones.inc" #include "woods.inc" global_settings { max_trace_level 5 } //Place the camera camera { sky <0,0,1> //Don't change this direction <-1,0,0> //Don't change this right <-4/3,0,0> //Don't change this location <10,10,10> //Camera location look_at <0,0,0> //Where camera is pointing angle 15 //Angle of the view--increase to see more, decrease to see less } global_settings { ambient_light White*2 } //Ambient light to "brighten up" darker pictures //Place a light--you can have more than one! light_source {<10,5,10> color White*2 } light_source {<10,-5,10> color White*2 } //Set a background color background { color VLightGrey } //The Icosahedron (adapted from Laura Berry) //Vertices are defined with tt so that the icosohedron is developed from an octahedron #macro myicosahedron(tt) //The 12 vertices #declare p1 = <tt,0,-1>; #declare p2 = <-tt,0,-1>; #declare p3 = <0,-1,tt>; #declare p4 = <0,-1,-tt>; #declare p5 = <-1,tt,0>; #declare p6 = <-1,-tt,0>; #declare p7 = <0,1,tt>; #declare p8 = <0,1,-tt>; #declare p9 = <1,tt,0>; #declare p10 = <1,-tt,0>; #declare p11 = <tt,0,1>; #declare p12 = <-tt,0,1>; //The 20 triangular faces #declare f1 = polygon { 4, p1, p8, p9, p1}; #declare f2 = polygon { 4, p1, p10, p9, p1}; #declare f3 = polygon { 4, p1, p10, p4, p1}; #declare f4 = polygon { 4, p1, p2, p8, p1}; #declare f5 = polygon { 4, p1, p2, p4, p1}; #declare f6 = polygon { 4, p12, p7, p11, p12}; #declare f7 = polygon { 4, p12, p7, p5, p12}; Platonic Solids and the Five-Intersecting Tetrahedra Page 16 of 21 Stephenie Swope #declare f8 = polygon { 4, p12, p5, p6, p12}; #declare f9 = polygon { 4, p12, p6, p3, p12}; #declare f10 = polygon { 4, p12, p3, p11, p12}; #declare f11 = polygon { 4, p3, p10, p11, p3}; #declare f12 = polygon { 4, p9, p10, p11, p9}; #declare f13 = polygon { 4, p7, p9, p11, p7}; #declare f14 = polygon { 4, p7, p8, p9, p7}; #declare f15 = polygon { 4, p5, p8, p7, p5}; #declare f16 = polygon { 4, p5, p2, p6, p5}; #declare f17 = polygon { 4, p2, p6, p4, p2}; #declare f18 = polygon { 4, p2, p5, p8, p2}; #declare f19 = polygon { 4, p3, p4, p10, p3}; #declare f20 = polygon { 4, p3, p4, p6, p3}; //Unite the faces to form the icosahedron object { union { object{f1} object{f2} object{f3} object{f4} object{f5} object{f6} object{f7} object{f8} object{f9} object{f10} object{f11} object{f12} object{f13} object{f14} object{f15} object{f16} object{f17} object{f18} object{f19} object{f20} } } #end //The Dodecahedron (adapted from the icosohedron using the center of each icosohedron face as a vertex #macro mydodecahedron(tt) #declare p1 = <tt,0,-1>; #declare p2 = <-tt,0,-1>; #declare p3 = <0,-1,tt>; #declare p4 = <0,-1,-tt>; #declare p5 = <-1,tt,0>; #declare p6 = <-1,-tt,0>; #declare p7 = <0,1,tt>; #declare p8 = <0,1,-tt>; #declare p9 = <1,tt,0>; #declare p10 = <1,-tt,0>; #declare p11 = <tt,0,1>; #declare p12 = <-tt,0,1>; //The 20 vertices #declare v1 = (p1+p8+p9)/3; #declare v2 = (p1+p9+p10)/3; #declare v3 = (p1+p4+p10)/3; #declare v4 = (p1+p2+p8)/3; #declare v5 = (p1+p2+p4)/3; #declare v6 = (p7+p11+p12)/3; #declare v7 = (p5+p7+p12)/3; #declare v8 = (p5+p6+p12)/3; #declare v9 = (p3+p6+p12)/3; #declare v10 = (p3+p11+p12)/3; #declare v11 = (p3+p10+p11)/3; #declare v12 = (p9+p10+p11)/3; #declare v13 = (p7+p9+p11)/3; #declare v14 = (p7+p8+p9)/3; #declare v15 = (p5+p7+p8)/3; #declare v16 = (p2+p5+p6)/3; #declare v17 = (p2+p4+p6)/3; #declare v18 = (p2+p5+p8)/3; Platonic Solids and the Five-Intersecting Tetrahedra Page 17 of 21 Stephenie Swope #declare v19 = (p3+p4+p10)/3; #declare v20 = (p3+p4+p6)/3; //The 12 pentagonal faces #declare fd1 = polygon { 6, v1, v4, v18, v15, v14, v1 }; #declare fd2 = polygon { 6, v1, v2, v3, v5, v4, v1 }; #declare fd3 = polygon { 6, v1, v2, v12, v13, v14, v1 }; #declare fd4 = polygon { 6, v2, v3, v19, v11, v12, v2 }; #declare fd5 = polygon { 6, v3, v5, v17, v20, v19, v3 }; #declare fd6 = polygon { 6, v4, v5, v17, v16, v18, v4 }; #declare fd7 = polygon { 6, v7, v8, v16, v18, v15, v7 }; #declare fd8 = polygon { 6, v6, v10, v11, v12, v13, v6 }; #declare fd9 = polygon { 6, v6, v7, v15, v14, v13, v6 }; #declare fd10 = polygon { 6, v6, v7, v8, v9, v10, v6 }; #declare fd11 = polygon { 6, v8, v9, v20, v17, v16, v8 }; #declare fd12 = polygon { 6, v9, v10, v11, v19, v20, v9 }; //Unite the faces to form the dodecahedron object { union { object{fd1} object{fd2} object{fd3} object{fd4} object{fd5} object{fd6} object{fd7} object{fd8} object{fd9} object{fd10} object{fd11} object{fd12} } } #end //The 5-Intersecting Tetrahedron (made by creating each tetrahedron from polygons made from specific vertices of the dodecahedron #declare tt=(sqrt(5)-1)/2; #declare p1 = <tt,0,-1>; #declare p2 = <-tt,0,-1>; #declare p3 = <0,-1,tt>; #declare p4 = <0,-1,-tt>; #declare p5 = <-1,tt,0>; #declare p6 = <-1,-tt,0>; #declare p7 = <0,1,tt>; #declare p8 = <0,1,-tt>; #declare p9 = <1,tt,0>; #declare p10 = <1,-tt,0>; #declare p11 = <tt,0,1>; #declare p12 = <-tt,0,1>; //The 20 vertices #declare v1 = (p1+p8+p9)/3; #declare v2 = (p1+p9+p10)/3; #declare v3 = (p1+p4+p10)/3; #declare v4 = (p1+p2+p8)/3; #declare v5 = (p1+p2+p4)/3; #declare v6 = (p7+p11+p12)/3; #declare v7 = (p5+p7+p12)/3; #declare v8 = (p5+p6+p12)/3; #declare v9 = (p3+p6+p12)/3; #declare v10 = (p3+p11+p12)/3; #declare v11 = (p3+p10+p11)/3; #declare v12 = (p9+p10+p11)/3; #declare v13 = (p7+p9+p11)/3; #declare v14 = (p7+p8+p9)/3; #declare v15 = (p5+p7+p8)/3; #declare v16 = (p2+p5+p6)/3; #declare v17 = (p2+p4+p6)/3; #declare v18 = (p2+p5+p8)/3; #declare v19 = (p3+p4+p10)/3; #declare v20 = (p3+p4+p6)/3; //Construct first tetrahedron by describing its polygons Platonic Solids and the Five-Intersecting Tetrahedra Page 20 of 21 Stephenie Swope #declare r=(sqrt(5)-1)/2; union{ object{myicosahedron(r) texture{pigment {color rgbf <s,1,s,s*f>}}} object{mydodecahedron(r) texture{pigment{color rgbf <1,0,0,0>}}} rotate<0,0,clock*rate> } #break #range(2.0001,3) #declare r=(sqrt(5)-1)/2; object{mydodecahedron(r) texture{pigment{color rgbf <1,0,0,0>}} rotate <0,0,clock*rate>} #break #range(3.0001,4) #declare s=clock-3; #declare r=(sqrt(5)-1)/2; union{ object{mydodecahedron(r) texture{pigment {color rgbf <1,s,s,s*f>}}} object{myonetetra texture{pigment{color Gold*2}}} rotate<0,0,clock*rate> } #break #range(4.0001,4.5) #declare r=(sqrt(5)-1)/2; object{mytwotetra texture{pigment{color Gold*2}} rotate <0,0,clock*rate>} #break #range(4.5001,5) #declare r=(sqrt(5)-1)/2; object{mythreetetra texture{pigment{color Gold*2}} rotate <0,0,clock*rate>} #break #range(5.0001,5.5) #declare r=(sqrt(5)-1)/2; object{myfourtetra texture{pigment{color Gold*2}} rotate <0,0,clock*rate>} #break #range(5.5001,6) #declare r=(sqrt(5)-1)/2; object{myfivetetra texture{pigment{color Gold*2}} rotate <0,0,clock*rate>} #break #range(6.0001,7) #declare r=(sqrt(5)-1)/2; union{ object{mydodecahedron(r) texture{pigment {color rgbf <1,1,1,0.7>}}} object{myfivetetra texture{pigment{color Gold*2}}} rotate<0,0,clock*rate> } #break #end iReferences: Most information taken from knowledge learned in University of Kentucky HON 301 Visualizing Mathematics Class: Notes located at http://www.ms.uky.edu/~lee/visual05/visual05.html Make Shapes Series No. 2: 8 mathematical models to cut out, glue and decorate, Gerald Jenkins and Anne Wild, Tarquin Productions 1990, Original Edition 1978, ISBN 0 906212 01 4 ii POV-Ray is a free Ray tracing software available at http://www.povray.org/ iii Taken from file developed by Dr. Lee published on http://www.ms.uky.edu/~lee/visual05/povray/povray.html