# Basic Principles of Mathematics and Geometry of Ritual Circles.

This treatise is an attempt to resolve the principles and theories that I have researched over the recent years. This is a work in progress and owes a lot to those who have assisted me and inspired me to expound upon the ancient lore contained here in.

The first part of this treatise deals with the numerological principles that apply to arcane architecture.

#### The Principle of Four

The prime elements are arranged in groupings of four, although they could be considered to be arranged in pairs, each pair only makes sense when associated with another pair. In addition the number four is of importance with regard to the mathematics of magic. The number four is considered the base number and when multiplied by itself we get sixteen, the mediant number of the system. To determine the base number taken to its further possible resonance we multiply the mediant number by itself to get two hundred and fifty six, which is called the closate number of the system.

With regard to ritual circles the resonance field of the ritual circle extends to the ninth iteration of the node measurements as described by Herodotus which is at the closate position, that is if each node iteration is measured in yards then the resonance field extends to two hundred and fifty six yards from the centre of the circle.

The iteration positions for the points of resonance related to a ritual circle lie at the following unit distances from the centre of the circle 1,2,4,8,16,32,64,128 and 256. Each of these positions are a multiple of two (strictly 1 cannot be arrived as a multiple of two but can be considered so as a special instance in this case).

#### The Principles of Contagion, Position and Orientation

These three Laws of Nostra Kari numerology when applied together explain the physical geometry of a ritual circle.

Position states that no two resonant nodes may reside within the same space, meaning that the elemental nodes within a ritual circle must be spaced out, and the best arrangement of such spacing is an even spacing.

Orientation states that each resonant (elemental) node is tied with a prime direction (north, south, east and west) and that the elemental nodes align with the following directions:

North – Water, South – Fire, East – Earth, West – Air

Contagion states that and ethereal entity is tainted by the presence of a physical entity which explains the positioning of the ethereal elements within the ritual circle. Each ethereal element lies between two physical elements

Void lies between Air and Water, Light between Water and Earth, Time between Earth and Fire, Darkness between Fire and Air.

The Law of Contagion also implies that we can infer the properties of an ethereal element by its associations with the physical elements that surround it.

#### The Principle of Shapes

The Nostra Kari numerology assigns a hierarchy to geometric shapes with regard to its affinity to magical architecture.

The square and the circle are considered the most sacred since they expound the balance of the elements. If one takes the square and places the eight manifest elements at the corners and midpoints of each side we find that they are equally spaced, fulfilling the most harmonious arrangement of the elements. The circle can also allow us to evenly space the elements around its circumference. There is also an interesting geometry of circles within squares or squares within circles. Considering these points it is not surprising that ritual circles are circular and the sinkholes are square. The second most important shape is the octagon, this can be constructed by joining the nodes spaced on a circle by straight lines. It can also be generated within a square by connecting a node to all other nodes except its opposite by a straight line. The shape formed at the centre of the square is an octagon. Finally because the manifest elements exist in two pairs of four it makes sense that an eight sided shape would have magical significance.

The least important shape is the rectangle, which shares some properties of a square and also can be formed by joining opposite faces of an octagon with straight lines.

From theses shapes above we can infer that simple regular shapes hold more magical affinity. Taking an extension from the above we can also infer that a sixteen sided shape also has magical affinity but due to its complexity has less of an affinity than those forms listed above.

#### Physical Geometry of Ritual Circles.

Noting the above principles we can construct a circular configuration of the manifest elements both ethereal and physical. We start with the manifest physical elements and their known orientation to each of the Prime Directions constructing the first grouping of four. Then each ethereal element is place between the physical elements as determined by their correspondences. The position of the elemental nodes is then placed equally spaced around the circumference of a circle. This geometric arrangement is what we see within an actual ritual circle and complies with the simplifications of the Concacatation diagrams postulated by Herodotus, Circero and others.

#### Mathematics of Ritual Circles

As stated above there is a pattern to the way numbers interact with ritual circles, and that the number two and most especially four are very important.

From Herodotus and others a system for measuring the position of the elemental nodes within the configuration of the ritual circle has been postulated. The most manual method is by the use of sense magic to determine “high points” within a ritual circles resonance field based on the assumption that these “high points” occur at regular predictable intervals processing away from the centre of the ritual circle.

Herodotus stated that the first node of resonance lay at approximately one eight the radius of the ritual circle from the centre of the circle (actually the documents state that it is at one eighth the diameter, but radius makes more sense). Using this as a base distance and then applying the use of the ability to Sense magic it is possible to determine a more precise location of the first node. Further the secondary, tertiary and finally the quaternary (or True node) are then located by multiplying the previous node distance by two and sensing that distance from the centre of the circle.

Each node would then lie at a given distance from the centre of the circle, dependant on the elemental balance of each circle.

The position of the nodes can be used to determine whether a particular elemental node lies outside of the circle itself indicating ‘leakage’ and possibly indicating an unstable ritual circle. Also the knowledge of the elemental balance within the circle can allow a ritualist to have more control over the energies within the circle.

Each ritual circle has a potential that determines the maximum amount of magical energy (measured in Thaums) that can safely be channelled by a ritualist through the circle. This potential directly relates to the size of the ritual circle and we can say that larger ritual circles have a greater Thaumic Potential.

To calculated the Thaumic Power of a ritual circle we must take the sum of the true node distances and divide the total by eight to get the prime potential of the circle. Take the diameter of the circle and multiply by two to get the enchantment vale height of the circle.

The enchantment vale height minus four multiplied by the prime potential gives the Total Circle Potential. To determine the Circle Threshold we need to apply Straudo’s Constant, however this factor of the equation is still unknown to me.

These calculations were used to determine the safety of an individual ritual circle during the times of the Empire and I suspect were involved with the fine tuning of a ritual circle to alter the characteristics of a ritual circle.

#### Magical Geometry of Ritual Circles

Apart from the physical structure and arrangement of a ritual circle, we also are aware of the magical forces that impinge on the circle. Some of this was outlined above with regard to the mathematics that apply to ritual circles, in that the points of resonant interaction within the ritual circle can be determined by a structured mathematical approach. The reason for this predictable pattern of interaction is due to the way that magical energy, also called void energy, flows away from the centre of the ritual circle. This flow manifests as waves that get longer the further they get away from the centre of the circle. These waves only relate to the manifest elements both physical and ethereal but the ideological (or unmanifest) elements are also integral to a ritual circle, and indeed a sinkhole. The Law of Orientation also states that the unmanifest elements must be tied to the manifest and hence must be involved in the magical structure of the ritual circle.

Extrapolating from the structure of a sinkhole, which focuses energy out of our world, which has both a manifest lens and an unmanifest lens we can say that a ritual circle must have an unmanifest component. In addition, since the unmanifest lens lies closest to the Void it makes sense that the unmanifest elements have an interaction that is closer to the Void than the manifest elements. From historical documents this interaction occurs halfway up the cone formed by the ritual seal and there are also waves of energy that flow out from this interaction across the resonance field of the ritual circle. This is represented in the Auriole displacement diagram which indicates that at the seventh and eighth iterations the manifest and unmanifest waveforms interact. The significance of this is unknown to me but there must be something important about these locations.

Cicero Albus postulated that the waves of void energy flowing out from the ritual circle swirl out in a spiral pattern away from the centre of the circle, rather than “marching away” from the middle of the circle as put forward by Herodotus. This addendum has merits as it is known that the void energies do indeed swirl down the ritual seal into the circle and that a swirling interrelation makes sense in the light of this. There is no effect on the mathematics of the ritual circle and the nodes of resonant interaction but it is a more accurate representation of the void energy flow within the circle.

#### Interaction of a Ritualist and Contributors within a Ritual Circle

Another important feature of ritual circles is to understand what happens when a ritualist draws upon the power of a ritual circle and how the void energy flows through him and his contributors.

When a ritualist enters a circle and seals it, he is opening the circle to the raw void energies “above” and drawing down the ritual seal around him containing this energy within. The ritualist acts as a focus for these void energies and is limited to the amount of energy he can contain by his natural talent. The majority of ritualists cannot perform rituals by themselves and require the assistance of contributors. These are individuals who have a limited ability to control void energy, this extends to containing and managing the void energy that the ritualist is calling on but cannot manage as limited by his natural talents. Each contributor is capable of managing an amount of void energy known as a Thaum, and it is known that a ritual requires a minimum of fifteen thaums to function, although there is a theoretical upper limit to the amount of energy that a ritualist can control even with the help of contributors and dependant on the properties and characteristics of the circle.

Void energy focussed through the ritualist is then used to perform acts of True Magic, manipulating the patterns within the circle by act of the ritualists will. It has been postulated that a ritualist subconsciously moves towards the elemental nodes most appropriate to the act that he is performing, but I find it more likely that the nodes temporarily align themselves to be in accordance with his will, in effect an “image” of the node moves.