Flatland to Sphereland & Beyond
Part I
©Lorena Loo
Part I
©Lorena Loo
There is an old adage about everything being just a matter of perspective. We are going to find out that is very true in going from 2 dimensions to 3 and vice versa in the realm of sacred geometry. In the process, a number of things heretofore not seen as integrally connected in sacred geometry will be unveiled. There was rhyme and reason to things the ancient Egyptians emphasized, though they left it as a mystery for others to figure out why.
Cubing the Square Roots
Three square roots were of foremost importance to the ancient Egyptians:
Cubing the Square Roots
Three square roots were of foremost importance to the ancient Egyptians:
Now we are rotating our cube so that we look squarely at a face of the cube as in the diagram to the left below. Through a simple construction, we can double the length of one of the edges of the cube face to get a length of two in the horizontal direction. We complete the other two sides of the rectangle and we have a golden rectangle with length to width ratio of phi!
© Copyright Lorena Loo
By forming the hypotenuse, we now have the square root of 5. All of this just falls right out of the cube. These most revered square roots by the ancient Egyptians just happen to inherently arise right out of the cube. Amazing. But we are by no means finished. There is more gold to be extracted from the cube.
Something Golden This Way Comes
The square root of 5 extracted this way within the cube allows us to push the envelope further. The diagram on the left below shows our three square roots right angle triangle constructed from the cube.
With one end of the compass point fixed at the vertex of the cube indicated by the green dot in the diagram, we can swing the other end to create an arc of radius the square root of 5. Where that arc intersects the extended cube edge (red dot) produces a line of length the square root of 5. Together with the edge length of the cube lying along the same line, we have a line of length
which is precisely !
Something Golden This Way Comes
The square root of 5 extracted this way within the cube allows us to push the envelope further. The diagram on the left below shows our three square roots right angle triangle constructed from the cube.
With one end of the compass point fixed at the vertex of the cube indicated by the green dot in the diagram, we can swing the other end to create an arc of radius the square root of 5. Where that arc intersects the extended cube edge (red dot) produces a line of length the square root of 5. Together with the edge length of the cube lying along the same line, we have a line of length
which is precisely !
All of these sacred geometric values are inherently derivable from the simple cube. One might say a gold mine of sacred geometry lay hidden in plain view, right in front of our eyes in the form of the cube. In order to see it, we needed to know a few things about sacred geometry and then be able to play with the cube to realize these things were inherently (with a little work) part of its system.
At the end of this article, we will return to this golden rectangle.
Cubing the Squares and Diagonalizing the Tetrahedron
A cube consists of 8 vertices and 6 equal-sized square faces and 12 edges. If we arrange 6 equal-sized squares in the form of a cross, we can easily fold up the squares to form a cube.
At the end of this article, we will return to this golden rectangle.
Cubing the Squares and Diagonalizing the Tetrahedron
A cube consists of 8 vertices and 6 equal-sized square faces and 12 edges. If we arrange 6 equal-sized squares in the form of a cross, we can easily fold up the squares to form a cube.
We now have the square root of three by virtue of the right angle triangle formed by the edge of the cube with the diagonal of the face and the diagonal of the cube.
The key to interrelating these three square roots lies in probably the simplest (to the average person) and most recognizable 3-dimensional geometric shape outside of the sphere. That is the cube.
At any vertex of the cube, three mutually orthogonal (perpendicular) edges of the cube meet. Those edges happen to form the axes of the Cartesian coordinate system in 3-dimensions.
At any vertex of the cube, three mutually orthogonal (perpendicular) edges of the cube meet. Those edges happen to form the axes of the Cartesian coordinate system in 3-dimensions.
If we had a cube of edge lengths s, the diagonal of any face of the cube would simply be s multiplied by the square root of 2.
Now we form the diagonal of the cube which is simply the line between any two opposite vertices of opposite faces. For example, if we start at a vertex in the bottom face of the cube, we must end up at the opposite vertex to it but on the top face of the cube.
Now we form the diagonal of the cube which is simply the line between any two opposite vertices of opposite faces. For example, if we start at a vertex in the bottom face of the cube, we must end up at the opposite vertex to it but on the top face of the cube.
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.
In sacred geometry, diagonals are said to be very important. We know the square root of 2 is the diagonal of a square with edges of unit length. If we form the diagonals of our cross made from 6 squares so the diagonals zigzag horizontally and vertically, something remarkable occurs when we fold it up into a cube.
Here we show the folding up of the "squares" cross into a solid cube. The diagonals of the cube fold in such a way as to form a tetrahedron perfectly nested inside the cube.
Here is another view of the tetrahedron nested in the cube. The tetrahedron has 4 equilateral triangular faces all of the same size, 6 edges and 4 vertices. Each of its 4 vertices lie at one of the 8 vertices of the cube. Each edge of the tetrahedron forms a diagonal of a square face of the cube and the cube has 6 faces.
We end up with two interpenetrating and opposite tetrahedrons which form a star tetrahedron. Tilting your head to the right will probably make the star tetrahedron more apparent to you in the image to the left above. The star tetrahedron is perfectly nested in the cube with each of its 8 vertices lying at one of the 8 vertices of the cube. Joining the vertices of the star tetrahedron will end up forming the cube within which it is nested.
The second image above is a photo of a model I built. I used a Zometool kit to build the cube but with a Zometool kit it is not possible to construct the diagonal of a square. So I used colored straw to do that, threading plastic coated wires through the straws and the zomeball connectors. You can see some of those wires dangling out from the connectors.
For those familiar with the Flower of Life teachings and the merkaba, the photo illustrates a "male" star tetrahedron or merkaba. To get a view of what the "female" star tetra or merkaba looks like, you have to rotate the view above away from you and "into" the page by 70.53º to two decimals (or
).
The second image above is a photo of a model I built. I used a Zometool kit to build the cube but with a Zometool kit it is not possible to construct the diagonal of a square. So I used colored straw to do that, threading plastic coated wires through the straws and the zomeball connectors. You can see some of those wires dangling out from the connectors.
For those familiar with the Flower of Life teachings and the merkaba, the photo illustrates a "male" star tetrahedron or merkaba. To get a view of what the "female" star tetra or merkaba looks like, you have to rotate the view above away from you and "into" the page by 70.53º to two decimals (or
).
edge lengths are x the edge length of the cube in which it is nested). One edge from each of the tetras intersect at the exact center of a face of the cube. There are 6 such intersection points and they will be key in constructing another Platonic solid nested within the same cube. We will get to that in the next section.
We have proceeded from 2-dimensional squares and their diagonals to a cube with its nested tetrahedron and star tetra. It may seem like going backwards, but now we are going to look at what our 3D objects appear as in 2D as projections in Part II.
Part II deals with Shadow Play and Inward Nesting of the Platonics.
We have proceeded from 2-dimensional squares and their diagonals to a cube with its nested tetrahedron and star tetra. It may seem like going backwards, but now we are going to look at what our 3D objects appear as in 2D as projections in Part II.
Part II deals with Shadow Play and Inward Nesting of the Platonics.
The cube is comprised of six equal-sized square faces and eight vertices. Take any square face and form the diagonal. The diagonal halves the square to form two right-angled triangles. The shared diagonal is the hypotenuse of each right-angled triangle. For a cube of edge lengths 1, the diagonal of any face of the cube is the square root of 2.
What would happen if we go back to our 6 squares arranged in the form of a cross and now form the other set of diagonals in the squares and then fold it up into a cube? We would expect to form another tetrahedron but because we have formed an opposite zigzag pattern of diagonals from the first set, we should form an tetrahedron opposite to the first. That is precisely what happens.
Now we have the square root of 3, we can set one end of the compass at the vertex indicated by the yellow dot and swing the other end to create an arc of radius square root of three to intersect the extended cube edge at the red dot. We now have a line length of the square root of three which happens to be perpendicular (at 90 degrees) to the square root of 2 diagonal.
The two diagonals connect to diagonally opposite edges of the cube. Both these diagonals lie in the same plane and in a square root of 2 rectangle shaded in gold to the left.
Later on, we will see how that rectangle becomes a square root of 3 rectangle from a different perspective and dimension.
One more square root to go and getting to that requires just a little thought.
Later on, we will see how that rectangle becomes a square root of 3 rectangle from a different perspective and dimension.
One more square root to go and getting to that requires just a little thought.
Purple line is diagonal of (square) face of cube. Orange line is diagonal of cube.
The photo to the left is what it looks like for the female star tetra but without the thumb. The thumb belongs to me. Holding onto the model with one hand was the only way I had to get it into the right position while I took a picture with my other hand. I also used a foot to hold in place one of the wires. Thankfully the foot did not make it into the photo as well.
Remember that the edges of the tetrahedrons are simply the diagonals of the cube's faces. So a tetrahedron nested in a cube of unit edge length has
edge lengths of (i.e. tetrahedron's
Remember that the edges of the tetrahedrons are simply the diagonals of the cube's faces. So a tetrahedron nested in a cube of unit edge length has
edge lengths of (i.e. tetrahedron's
