12 cubeoctahedral spheres surround the central sphere. The centres of the spheres are the same distance from their neighbors as they are from the centre of the central sphere. The cubeoctahedron is the only shape in 3d that does this, and for this reason Buckminster Fuller called it the Vector Equilibrium. At 19mm diameter this is our smallest version of this form. This demonstrates the natural way that spheres cluster. In the same way that 6 circles perfectly surround a central circle 12 spheres surround a central sphere. It is the most basic example of the tetrahedral symmetry of space itself...
This is the standard two dimensional drawing of the 4 dimensional cube known as the hypercube or tesseract. A technical description can be found on Wikipedia here https://en.wikipedia.org/wiki/Hypercube
This design has all 8 cubes identically depicted. Each of the 8 sides of the bounding octagon is the side of a square.
Many variations on our unique three dimensionalization of the hypercube can be found here
NOTE: the colored 'sandstone' is a grainy slightly soft plastic and is not color fast. (Dont leave it in hot sun or get it wet). The StarTetrahedron is one of the most important Sacred Geometrical forms. The tetrahedron is the simplest of the 5 Platonic Solids, but it is not fully symmetrical. However when two of them are interpenetrating like this then they are fully balanced, and their outer vertexes form a cube whilst their inner vertexes form an octahedron. Thus In one form the StarTetrahedron contains the 3 core shapes that exist in all dimensions, the 'Simplex', the 'Cross'...
Here is what crystal books have to say about Aventurine. Aventurine is known as the "Rock of Chance," thought to be the luckiest of all crystals, particularly in manifesting prosperity, or for enhancing your luck in competitions or games of chance. Its winning power makes it a fantastic ally for boosting one's chances in any kind of scenario - an initial date, tax obligation audit, also landing a promotion. One requires only to be near it to obtain its benefits.This stunning stone, however, is not just an attractor of luck. Aventurine releases old patterns, and frustrations so brand-new growth can...
This is one of my studies in 3 dimensionalizing the tradition Metatron's Cube design. Each 'shell' has 12 spheres just as the traditional design has 6 circles touching the innermost circle. If the centre of each of these 24 spheres were joined to every other point we would have an exact 3D analog of the 2D Metatron's Cube. By analogy then if the 2D form is to be known as Metatron's Cube, then this form must be one dimension higher and be known as Metatron's Hypercube This form is provided in a variety of colors in resin as metals will...
This is the seed that gives rise to the tree that blooms into the flower of life. It is no accident that from some angles it looks like a child's drawing of an atom! The 12 outer vetexes of this Ganesha sphere are the points of the cubeoctahedron, also known as the vector equilibrium. The sphere with 6 axes displays the triangular and hexagonal symmetries of the isotropic vector matrix. This is the most fundamental geometry of space, of particles and atoms, electrical fields, and cosmological bodies. These same symmetries give rise to the double torus through rotation producing an equatorial plane...
This form is one of the simplest expressions of the fundamental symmetries of 3d space. It is composed of 4 circles that intersect at 12 points. The 6 axis show the hexagonal symmetry that Buckminster Fuller proposed is more fundamental than the cubic symmetry that obsesses Western culture. It is no accident that from the right angle the form appears exactly like a child's drawing of an atom.
Larger 30mm version available here
This form gives an amazing impression of dynamic movement. Like my Toroidal Hypercube it seems to spin in both directions at the same time.All the higher dimensional forms stretch the mind to try and understand their higher dimensional logic. We intuitively recognise that it makes less sense as a 3 dimensional form than as a projection of something simpler operating in a more expansive space. I believe that this form is one of the 4th dimensional 'Archimedian hypersolids' as found in a lecture by one of the 20th centuries pre-emminent geometers H.S.M. Coxeter. Version with hanging ring for pendant available...
This form gives an amazing impression of dynamic movement. Like my Toroidal Hypercube it seems to spin in both directions at the same time. Like all the higher dimensional forms it stretches the mind to contemplate its higher dimensional login. We intuitively recognise that it does makes less sense as a 3 dimensional form than as a projection of something simpler operating in a more expansive space. I believe that this form is one of the 4th dimensional 'Archimedian hypersolids' as found in a lecture by H.S.M. Coxeter. A version without the hanging ring is available here
This is the 32mm version. We also have a version with a hanging ring here and a larger 50mm version here 'Metatron's Compass' is a unique geometry in many ways. It can also be named 'the 4 Dimensional Vector Equilibrium', 'the 4 dimensional CubeOctahedron', or 'the Toroidal projection of the 24 Cell Polytope', but I prefer to call it Metatron's Compass as I believe it is the secret behind the popular 2 dimensional design known as Metatron's Cube. The importance of Metatron's Cube lies in its unification of the various possible symmetries of space, as shown in the way that the...
This is the 50mm version. We also have a smaller 32mm version here 'Metatron's Compass' is a unique geometry in many ways. It can also be named 'the 4 Dimensional Vector Equilibrium', 'the 4 dimensional CubeOctahedron', or 'the Toroidal projection of the 24 Cell Polytope', but I prefer to call it Metatron's Compass as I believe it is the secret behind the popular 2 dimensional design known as Metatron's Cube. The importance of Metatron's Cube lies in its unification of the various possible symmetries of space, as shown in the way that the outlines of the 5 Platonic solids can be...
This is a never before seen projection of the 5 dimensional cube into 3 dimensions. What is a 5 dimensional cube? Glad you asked :-) Well there are many ways to try and understand what the geometers of the 19 century figured out about higher dimensional geometry. One of the easiest (though not necessarilly the most accurate) ways is by analogy to the dimensions that we are familiar with. In the third dimension we are familiar with the Cube, and we can see that it has similarities with the two dimensional Square. By moving a square at right angles to...
There is a 33% larger version of this here This is a never before seen projection of the 5 dimensional cube into 3 dimensions. I love the way simply by its 5 dimensional symmetries it transcends the duality between the dodecahedral and cubic forms in 3D space. I also love the way it 'folds up' so that all the lines remain identical lengths. It is based on my 4D toroidal hypercube model that keeps all 8 cubes equally distorted by the projection down into 3D. Here instead of 8 cubes we have 10 hypercubes equally distorted. I take no responsibility...
The Platonic Solids are so important to geometry that I thought to make them as affordable as possible. All 5 are the same approx 35mm height. The best way to understand these geometries is as symmetrical packing of spheres. If you have 4 identical spheres and squash them together, their centers will naturally arrange themselves in the shape of a tetrahedron, the simplest straight edged 3 dimensional form. Note that the distances and angles between all vertexes is identical. Perhaps a little less obvious (and less stable), 6 spheres will form an Octahedron. The more spheres you add the...
This is a never before seen projection of the 5 dimensional cube into 3 dimensions. What is a 5 dimensional cube? Glad you asked :-) Well there are many ways to try and understand what the geometers of the 19 century figured out about higher dimensional geometry. One of the easiest (though not necessarily the most accurate) ways is by analogy to the dimensions that we are familiar with. In the third dimension we are familiar with the Cube, and we can see that it has similarities with the two dimensional Square. By moving a square at right angles to...
This is a beautifully delicate version of the 64 Tetrahedron Grid that forms a key component in the physics of researcher Nassim Haramein. Many more variants can be found here
