A torus is defined by two parameters: the radius of the torus (that is the radius of the torus's defining circle, measured from the origin) and the radius of the tube (the perpendicular distance from the defining circle to the surface of the torus). These are R and rrespectively. Normally R > r.

An implicit definition of the torus is:

Toroidal Space is the name used to describe the area and volume of a torus or so-called doughnut shape. This special form has been used to describe and/or represent a number of things in our "real" actual material world, as well as, our "imaginary" potential one.

A major branch of geometry is the study of geometrical structures on manifolds. A manifold is a curved space of some dimension. For example, the surface of a sphere, and the torus (the surface of a doughnut), are both 2-dimensional manifolds. Here the manifold itself is only the background for some mathematical object defined upon it, as a canvas is the background for an oil painting. This kind of geometry, although very abstract, is closer to the real world than you might think. Einstein's theory of General Relativity describes the Universe - the whole of space and time - as a 4-dimensional manifold.

Space itself is not flat, but curved. The curvature of space is responsible for gravity, and at a black hole space and time are so curved they get knotted up. Everything in the universe - light, subatomic particles, pizzas, yourself - is described in terms of a geometrical structure on the space-time 4-manifold. Manifolds are used to understand the large-scale structure of the Universe in cosmology, and the theory of relativity introduced the idea of matter-energy equivalence, which led to nuclear power, and the atomic bomb. The universe is now considered to be a 12 dimensional nested manifold.

In "superstring" physics, the torus is known as the "perfect" shape. It is now accepted as a mathematical model that can be used to describe objects in space. Surface topology is superior to geometry for describing such phenomenon because it deals with much more sophisticated and profound spacial and temporal relationships.

Other scientists have also suggested that the entire universe may be shaped like a torus. Itzak Bentov in a final book called A Brief Tour of HIgher Consciousness published just after his passing is dedicated to the idea that all reality including consciousness itself can be modeled with the toroidal manifold. His diagrams show galaxies as being toroidal forms with "white holes" putting out energy while "black holes" on the opposite side taking it back in. The inventor of the Bell Helicopter and other major innovations, Arthur Young, also explores the function of toroids as models of primary cosmology in his famous work...The Reflexive Universe.

For an image of the Earth’s field, which science refers to as the Van Allen Belt, click on and click on Enter the Site, then Our Planet, then Image Archives. See how the field is donut shaped close to the Earth? This is referred to in sacred geometry as the tube torus. It’s a shape that occurs often in Nature.

In particle physics, the doughnut shape is also known to provide the best environment within which to accelerate particles as it can hold and route the plasma formed by such machines in a more efficient manner. Toroidal geometry is attractive for space energy storage magnets because it results in small external magnetic fields. Russia first manifested this idea in their Tokamak accelerator. The new European Union and the United States have recently improved on the Russian model and are using these new designs to perform important experiments in plasma physics. These devices operate on the principle of fusion...the technique used by the sun and stars to produce their vast amounts of light and heat. One of the first and most important practical uses of such incredible devices will be to generate electrical power. All of the electricity required annually for one person can be produced by the fusion of only 500 liters of water!


When it comes to designing gravitational environmental habitations, the torus has been realized as the preferred shape within which to enclose a space station or star colony. The design below was engineered at Stanford university and is called the Stanford Torus. It is the most efficient way to distribute the space and achieve the gravity required to maintain life.


The toroidal shape has also been discovered by to be the most efficient way to wind an electrical transformer, as a coil wound in this configuration produces very clean, highly accurate, precise and reliable power. For that reason, toroidal transformers are employed by engineers in high-end audio equipment and other electronics to lower distortion and increase power at the same time.



In addition to its real world correlations, the torus has also been used to illustrate certain concepts of the so-called "subtle energy" worlds. The toroidal form figures heavily into the esoteric study of sacred geometry...a meta-science that reveals how shape and form are primary underlying principles of manifestation.

Since ancient times, "seers" have confirmed that the human aura appears as a series of nested spherical torus formations. Modern seers such as Barbara Ann Brennan (Hands of Light) have confirmed that each “chakra” is shaped like a two-ended trumpet, and this is a visualization of where each spherical torus or energy body has its axis. She describes the torus as the perfect shape for a black hole to white hole vortex, and also comments on the gyroscopic properties of this energy form. Ancient seers also went out-of-body and reported their observations of the structure of the CU, often calling it the “World Tree” after their sighting of the main central axis. We will show that this concept reappears in a remarkably similar fashion across a huge number of different cultures, and a clear connection to the nested spherical torus can easily be established; it is, literally, one and the same formation.


For many years esoteric researchers have been searching for the meaning of the Language of Light. Scholars of every school have wondered of this parabolic statement. Is there really such a thing as the Language of Light? The answer seems to be yes. There is contained within the structure of Light such a language. In one of the basic structures of physics, called the Tube Torus there is a study of vortex energy that describes Light and Language in it’s potential.


The Tube Torus is derived from the basic study of the Flower of Life. By using a simple compass one can create the Flower of life very easily. There are seven basic steps in the composition of the Seed of Life which, if continued to its conclusion, yields the study of the vortex through the Flower of Life.

One of the energy patterns around the body is in the shape of a torus - or donut - with the energy flowing through the body and looping around to connect in at the feet and the head. It is as though we are in the middle position of the donut. The flow in through the head and feet is bidirectional. In other words it flows like the tides one way and then the next. In the matrix of this flow are the wave and particle relationships that structure and govern the nurturing "cosmic" energy from which we are crystallized.

It has even been suggested that the torus can be used to define the workings of consciousness itself. In other words...consciousness has a geometry! The geometric shape used to describe the self-reflexive nature of consciousness is the torus. The torus allows a vortex of energy to form which bends back along itself and re-enters itself. It 'inside-outs', continuously flowing back into itself. Thus the energy of a torus is continually refreshing itself, continually influencing itself. Some amazing work has been done over the years by Dan Winter (whose books are now out of print), who suggests with geometric diagrams that the only way self-reflexiveness can continue without breaking down is by using waveforms based on harmonics of the golden section, which we have talked about in the Geometry section of this website. He claims our very DNA is a helix formed by 2 rotating geometric figures called icosahedrons and dodecahedrons. Stan Tenen (MERU Foundation) has also done some great work along these lines, contradicting Winter's work and basing his own discoveries on the work of Arthur Young, who also worked with the torus. Regardless of who is correct, the torus does seem to be the common denominator, the self-organizing way consciousness is designed.

The torus enables vortexes to be formed. All torii create energy vortexes. Energy can funnel up or down through it; the rotating nature of a torus generates a flow of energy through the torus, depending upon the speed of rotation of the torus and the kind of torus it is (there are 3 kinds of standard torii). An examination of the torus shows that its very construction forms energy funnels, or vortexes.


Mr. Winter and others has also shown how the biophysical manifestations such as the human heart and DNA have toroidal configurations. For more information contact the San Gral School of Sacred Geometry.



The toroidal shape used in HRT is similar to a doughnut but rather than having an empty central "hole", the topology of a torus folds in upon itself and all points along its surface converve together into a zero-dimensional point at the center called the Vertex. This makes it the perfect environment within which to populate the automatous partials that make up the essence of any dynamic function such as a person's vibrational essence. Any input placed at the Vertex while the torus is "torsioned" (folded and rotated inward) is spread out and distributed over the entire surface of the toroid (see below).


The spectral essence matrices produced in the HRT process are gathered up within such a toroidal manifold. A pure, coherent harmonic source is fed through the "portacle" - the modulating filter generated from the individual's voicing - and focused to a zero-dimensional point at the vortex thread (or "origin") of the flowing torus. As the torus rotates and folds through itself - each point passing through the central thread - its framework is populated by the partials of the spectral essence.

The torus, however, is not the most practical format for presentation, as the partials populating the interior or back-facing surfaces would be obscured from view. Unfolding the native toroidal space and projecting it into a 2-dimensional matrix allows us to see every partial in a uniform grid. The illustration above shows the dynamics of this process, and the characters to the right of the diagram is the actual unfolding algorithm. After unfolding is complete, the partials in the matrix are shifted by 1/2, thus restoring the origin to the center.

The unfolded torus produces a flat 256 x 256 pixel image that is sort of a cross-sectional view of the original torus. This image can then be viewed on a computer screen as biofeedback representing a person's spectral domain essence. As the torus is rotated through 360 degrees of the Chromaphase cycle, the unfolded image (called a Matrix) continually changes over time, providing something similar to a dynamic "Rorschach" (inkblot) image. Viewing this image allows one to take apart, re-arrange, re-distribute, and then re-assemble themself.