Cosmic Heart Help

Introduction

A polygon is a 2-dimensional figure with a certain number of sides. A square is a polygon with 4 sides.

A regular polygon is a polygon whose sides are of equal length and whose vertex angles are equal.

Polyhedra are 3-dimensional surfaces composed of polygons such that each edge is adjacent to two polygons, or faces. Regular polyhedra are those polyhedra that are composed of only one type of regular polygon.

There are five different convex regular polyhedra, also called the Platonic Solids after the Greek philosopher Plato who studied them. They are the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. See figure A.

Figure A

The following chart shows the number of faces, edges, and vertices for each of the 5 regular polyhedra. The number of faces, edges, and vertices for each polyhedron follow a relationship that is called Euler's Formula and which states that: faces - edges + vertices = 2.

There are many relationships that exist between these polyhedra as a consequence of Euler's Formula. One fascinating relationship is that a tetrahedron can be inscribed in a cube. That is, a tetrahedron of the proper size can be placed inside of a cube of the proper size so that there is a one-to-one relationship between the faces of the cube and edges of the tetrahedron; each of the 6 edges of the tetrahedron shows up as one of the diagonals of one of the 6 faces of the cube. Furthermore, the four vertices of the tetrahedron correspond to 4 of the 8 vertices of the cube. See figure B. Note that there are two different ways in which the tetrahedron can be inscribed in a cube since there are two ways in which the tetrahedron’s 4 vertices can match up with 4 of the 8 cube vertices.

Figure B

Similarly, a cube can be inscribed in a dodecahedron. Each of the 12 edges of the cube is a diagonal of one of the 12 pentagonal faces of the dodecahedron, and the 8 vertices of the cube correspond to 8 of the 20 vertices of the dodecahedron. See figure C. Note that there are 5 ways in which the cube’s 8 vertices can match up with 8 of the dodecahedron’s 20 vertices.

Figure C

Cosmic Heart Application

This application demonstrates the relationships outlined above wherein the tetrahedron can be inscribed in a cube in two different ways and a cube can be inscribed in a dodecahedron in 5 different ways. Furthermore, it demonstrates a kind of quantum fluctuation or “beating” as a heart does, in which the tetrahedron oscillates back and forth between the two positions in which it can be inscribed in the cube, and in which the cube oscillates randomly between the 5 positions in which it can be inscribed in the dodecahedron. These two oscillations can be independent, and although the duration of time which it takes for a quantum fluctuation to occur for the tetrahedron-cube pair need not be the same as the duration of time which it takes for a quantum fluctuation to occur for the cube-dodecahedron pair, in this application these two durations of time are the same. Therefore, when all three polyhedra are visible and they are fluctuating, the tetrahedron will always complete an oscillation within the cube at the same time that the cube completes an oscillation with the dodecahedron. This is more aesthetically pleasing than if the two fluctuation time frames were not synchronized.

Cosmic Heart Controls

Mouse and keyboard

• The model can be rotated about by clicking and holding the left mouse button and the moving the mouse around.
• You can zoom in and out by clicking and holding the right mouse button and then moving the mouse up and down, or alternatively, you can use the Insert and Delete keys on the keyboard to do zooming.

File menu

• Image files of types BMP, JPEG, and PNG can be created by selecting ‘Save image’ from the File menu.
• Images can printed by selecting ‘Print Preview’, or simply ‘Print’ from the File menu.
• AVI movies can be created by selecting ‘Save movie’ from the File menu.
• The size of the images and movies, as well as the number of frames in the movie, can be set in a dialog accessed by selecting ‘Set image size / movie frames’ from the File menu.

NOTE: When a movie is created, the rendering is not done to the screen so you will not see the frames which are being captured. Furthermore, the application will become frozen and unresponsive until frames have all been captured. Be patient. At this point a window should appear in which you can select the type of video compression to use when assembling the AVI movie from the frames which were captured. I recommend choosing ‘Microsoft Video 1’. The movie which is produced will be named ‘CosmicHeart.avi’ in the ‘movie’ directory within the folder which contains the ComsicHeart executable.

Control panel

All other controls are located in the control panel at the right of the screen.

At the top of the control panel is a box, labeled ‘REGULAR POLYHEDRA’ containing a smaller box for each of the three polyhedra ‘TETRAHEDRON’, ‘CUBE’, and ‘DODECAHEDRON’. See figure D. The visibility of each polyhedron can be toggled by clicking in the checkbox labeled ‘Visible’. Clicking on the buttons labeled ‘Position 1’, ‘Position 2’, etc, for the tetrahedron and the cube will orient that polyhedron within the cube or dodecahedron respectively in one of its possible inscribed positions. Also within this box is a control that will make only one particular face of each polyhedron visible. This was really just for purposes of testing, but I left the feature in place anyway.

Figure D

Beneath the box labeled ‘POLYHEDRA’ is another box labeled ‘Inscribing spheres’. See figure E. Because these regular polyhedra are composed of regular polygons, 3 spheres can be drawn for each of the regular polyhedra, which share their centers with the the center of each polyhedron and which are tangent to each of the polyhedra at the center of each face, or at the midpoint of each edge, or at each vertex of the polyhedron. The visibility of each of these three circumscribing spheres can be toggled using the checkboxes in the ‘Inscribing spheres’ box. Whether the spheres are rendered solid or as meshes can be toggled with the ‘Mesh’ checkbox.

Figure E

Beneath the box labeled ‘Inscribing spheres’ is another box labeled ‘Fluctuation’ which contains those controls determining the pattern of fluctuation of the tetrahedron within the cube and the fluctuation of the cube within the dodecahedron. See figure F. The first checkbox toggles whether or not the polyhedra are fluctuating. At least two of the three polyhedra must be visible in order to see fluctuation. When the polyhedra are fluctuating, each of the tetrahedron and the cube is rotating about a particular axis that passes through the center of one of the cube’s faces or the dodecahedron’s faces respectively. The axes of rotation do not change, unless the second checkbox labeled ‘Fluctuation is about random axes’ is checked. If it is checked, then after each quantum oscillation of the tetrahedron and the cube, the axis of rotation for the tetrahedron and for the cube is randomly selected. The last checkbox toggles the visibility of the rotation axes. The last control in the ‘Fluctuation’ box is a slider control which determines the speed or accuracy of the fluctuation. This control is only functional when the ‘fluctuation’ checkbox is checked and at least two of the three polyhedra are visible.

Figure F

At the very bottom of the control panel are two checkboxes. The first toggles the transparency of the polyhedra. See figure G. The second is meant to be used with standard 3D red-blue glasses.

Figure G

Polyhedra links

• Pavilion of Polyhedreality
• Virtual Polyhedra
• The Geometry Junkyard
• Uniform Polyhedra
• Polyhedra Pastimes
• Polyhedra Puzzles
  

Hope you enjoy this software.

Any thoughts, comments, suggestions, criticisms can be directed to me at brian_diloreto@yahoo.com

Thanks.

Brian DiLoreto 1/30/2002