Flatland to Sphereland & Beyond
Part II
©Lorena Loo
Part II
©Lorena Loo
Shadow Play
You may have noticed how the cube is tilted differently in the two views of the star tetrahedron for the male and female. If we looked at the view directly on the vertex that corresponds to the apex of the top (sun) tetrahedron, here is how the cube looks like.
You may have noticed how the cube is tilted differently in the two views of the star tetrahedron for the male and female. If we looked at the view directly on the vertex that corresponds to the apex of the top (sun) tetrahedron, here is how the cube looks like.
© Copyright Lorena Loo
Inward Nesting of the Platonics
We have found how amazingly simple it is to nest the tetrahedron (and star tetra) in the cube. What about the other Platonic solids? Let's find out.
We have found how amazingly simple it is to nest the tetrahedron (and star tetra) in the cube. What about the other Platonic solids? Let's find out.
Text and images (unless otherwise indicated in the credits) are copyrighted
© by Lorena Loo
In the instances where images in the public domain have been modified as in the case of geometrizing them, the modified images are no longer public domain but the copyright of the author who made the modifications. Here that means me, Lorena Loo. That is by copyright law.
© by Lorena Loo
In the instances where images in the public domain have been modified as in the case of geometrizing them, the modified images are no longer public domain but the copyright of the author who made the modifications. Here that means me, Lorena Loo. That is by copyright law.
Remember the square root of 2 rectangle we talked about in Part I? Well you can clearly see that rectangle in both views of the cube above. The difference is the left view has the top length of the square root of 2 rectangle tilted slightly closer to us than the bottom length. In the right image, the top length of that rectangle is not only further away from us than the bottom length, but the difference is greater.
What you see is a two dimensional photo of 3-dimensional objects. We end up with getting a perspective view which depicts objects or parts of an object which are further away from us as smaller and objects or parts of objects which are closer to us as larger. In this case, lines closer to us appear longer in length, lines further away appear shorter in length. In reality, all edges of the cube are equal in length.
If we shine a light directly at the vertex we are looking right at, the shadow projection would be very much like the images of the cube we see in the photographs above. This I know from experience as I have done this.
Perspective views are what we get through shining a light from a finite distance away. If we shine a light from an infinite distance away, we get an orthographic projection which does not take depth into account. That means if two lines are the same length, they are shown or seen as that even if one is at a considerable distance further away from us than the other.
What you see is a two dimensional photo of 3-dimensional objects. We end up with getting a perspective view which depicts objects or parts of an object which are further away from us as smaller and objects or parts of objects which are closer to us as larger. In this case, lines closer to us appear longer in length, lines further away appear shorter in length. In reality, all edges of the cube are equal in length.
If we shine a light directly at the vertex we are looking right at, the shadow projection would be very much like the images of the cube we see in the photographs above. This I know from experience as I have done this.
Perspective views are what we get through shining a light from a finite distance away. If we shine a light from an infinite distance away, we get an orthographic projection which does not take depth into account. That means if two lines are the same length, they are shown or seen as that even if one is at a considerable distance further away from us than the other.
An orthographic projection of either of the above cube views produces a regular hexagon (first image to left). All six sides of the hexagon are equal in length and it is divided into six equal pieces of "pie." The other image highlights the square root of 2
rectangle from the cube as an orthographic projection. Here it becomes a square root of 3 rectangle as the rectangle within a regular hexagon. Just a matter of perspective, a matter of dimension.
Now let's look at the two cube views with the star tetrahedrons nested within.
Now let's look at the two cube views with the star tetrahedrons nested within.
You can see how the lines of the top face of the bottom (earth) tetrahedron which are green in color appear longer in length of those of the bottom face of the top (sun) tetrahedron which are bright purple in color. Again this is a case of one face being closer to us than another and a shadow or perspective projection would depict that depth difference as a difference in line lengths even though in reality the edges of both tetrahedrons are the same lengths.
An orthographic projection would depict the lines as being equal in length.
An orthographic projection would depict the lines as being equal in length.
This is what it looks like. A regular hexagon divided into six equal pieces of pie with a hexagram (Star of David as it is also known as) perfectly embedded within. A perspective or normal shadow projection would not yield this. So if you have been told the hexagram is the projection of the star tetrahedron onto the 2-dimensional plane, you were not given the correct picture. That is true only with the orthographic projection or projection with a light source an infinite distance away.
As an orthographic projection of the cube-nested star tetra, the Flower of Life also easily lends itself to constructing a golden rectangle. Without going into all the details, I just present the diagram below which is similar to what I did with the cube in the section on Something Golden This Way Comes. This also applies to the orthographic projection of the two views of the cube in the photos at the top of this page.
Let's return to the cube with its nested star tetra. Recall the two diagonals of each square face of the cube intersect at the exact center of the face. By joining the centers of the four "side" faces of the cube sequentially, we form a square. By joining the centers of the top and bottom faces of the cube to the centers of the 4 side faces of the cube, we end up with an octahedron.
The animation to the left demonstrates what I mean.
The animation to the left demonstrates what I mean.
An octahedron has 8 equilateral triangular faces, 6 vertices and 12 edges. It can be thought of as two equal-sized pyramids stuck together so that they share the same square base. In fact, viewed from directly above any of its vertices, the octahedron looks like a pyramid with a square base and 4 equilateral triangular faces. It does not matter which vertex you look directly down upon the octahedron, the view is the same. That is also true of the tetrahedron viewed from directly above any of its vertices.
Next we look at the icosahedron nested within the cube. It will be a little bit trickier but first let's see what an icosahedron looks like. The photo to the right illustrates two different views of the icosahedron I made from a Zometool kit. An icosahedron has 20 equilateral triangular faces, 12 vertices and 30 edges.
We can easily determine the edge lengths of the nested octahedron in a cube of edge length 1 are . That is exactly half of the
edge lengths of the tetrahedron nested inside the same cube. In fact, if we add 4 tetrahedrons of edge lengths to 4 of the
octahedron's faces in a certain arrangement, we get the octahedron nested within the tetrahedron. Add 4 other tetrahedrons of the same edge lengths to the remaining 4 faces of the octahedron and we get the octahedron nested within the star tetra. Viewed from another perspective, the star tetra with its nested octa is like an octa nested within a cube.
edge lengths of the tetrahedron nested inside the same cube. In fact, if we add 4 tetrahedrons of edge lengths to 4 of the
octahedron's faces in a certain arrangement, we get the octahedron nested within the tetrahedron. Add 4 other tetrahedrons of the same edge lengths to the remaining 4 faces of the octahedron and we get the octahedron nested within the star tetra. Viewed from another perspective, the star tetra with its nested octa is like an octa nested within a cube.
Forming the nested icosahedron involves
forming lines of length that of the cube's edge lengths.
Since we are using unit edge lengths for the
cube, we must form lines of length . In the
diagram to the left, we want to form the brown line versus the cube's edge length represented by the blue line. If you are familiar with a phi cut of a line, it is the longer line segment of a phi cut of the cube's edge length that we want. Six such lines are formed, one on each face of the cube. Each line is centrally positioned on the face with lines on opposite faces parallel to one another. Don't worry about trying to absorb all I just said. If you read it and look at the animation to the left, you will comprehend it from the combination of the explanation and the visualization.
forming lines of length that of the cube's edge lengths.
Since we are using unit edge lengths for the
cube, we must form lines of length . In the
diagram to the left, we want to form the brown line versus the cube's edge length represented by the blue line. If you are familiar with a phi cut of a line, it is the longer line segment of a phi cut of the cube's edge length that we want. Six such lines are formed, one on each face of the cube. Each line is centrally positioned on the face with lines on opposite faces parallel to one another. Don't worry about trying to absorb all I just said. If you read it and look at the animation to the left, you will comprehend it from the combination of the explanation and the visualization.
So we have 6 brown lines from 3 sets of parallel lines. The last image in the animation sequence shows three of the lines, one from each set of the parallel lines, are perpendicular to each other like the three edges of a cube meeting at a vertex.
The red dots in the animation simply represent the endpoints of the brown lines. There is a reason they are there. Those red dots will be the 12 vertices of the nested icosahedron within the cube.
Now the fun begins. How do we form our icosahedron from this?
The red dots in the animation simply represent the endpoints of the brown lines. There is a reason they are there. Those red dots will be the 12 vertices of the nested icosahedron within the cube.
Now the fun begins. How do we form our icosahedron from this?
Cube view from directly on vertex. Left is for male star tetra, right is for female star tetra.
Left photo is with the star tetra oriented for males, right is for star tetra oriented for females.
Above left: Flower of Life symbol highlighting the orthographic projection of the cube and nested star tetra. Above right: Animation demonstrating how the vertices (points) of the star tetra form a cube.
You may recognize this orthographic projection as simply the Flower of Life or at least formed from the Flower of Life symbol.
Three different views of the nested octahedron. From left to right: Cube with nested tetrahedron and octahedron; cube with nested octahedron; tetrahedron with nested octahedron.
Now suppose we further subdivide those 4 smaller equilateral triangles exactly the same way we did with the big one out of which they were formed. This second order subdivision of the original triangle we represent by orange lines in the image to the right above. Essentially we've made a fractal or a repeating geometric pattern. Now if we fold the figure up again along the 3 innermost purple lines like we did before, we get a tetrahedron but this time the orange lines will connect together to form the octahedron nested within the tetrahedron!Each of the 6 vertices of the octa lie at the midpoint of one of the 6 edges of the tetra.
Had we started out with an equilateral triangle of edge lengths twice the square root of 2, the resulting tetrahedron and nested octa would be precisely that nested within a cube of unit edge length.
Creating such a nested 3D object from a 2D net is best done on a transparent sheet such as the ones you can buy at an office supply store or better yet, dollar stores. I came up with the idea for this net late one night while trying to think of the best way to visualize the octa nested within the tetra. If you do this, especially with a transparent sheet, you will see what I mean.
Had we started out with an equilateral triangle of edge lengths twice the square root of 2, the resulting tetrahedron and nested octa would be precisely that nested within a cube of unit edge length.
Creating such a nested 3D object from a 2D net is best done on a transparent sheet such as the ones you can buy at an office supply store or better yet, dollar stores. I came up with the idea for this net late one night while trying to think of the best way to visualize the octa nested within the tetra. If you do this, especially with a transparent sheet, you will see what I mean.
One last thing on the nested octahedron. Take an equilateral triangle and join the midpoints of all three sides to subdivide it into 4 equal-sized smaller triangles (first image to left). These 4 smaller triangles will also be equilateral (all 3 sides of equal length). If we fold this up along the 3 inner lines, we can form a tetrahedron.
