Natural Coordinates

Ray Whitmer

Contents

• 1 Introduction
• 2 Background and Acknowledgements
• 3 Uses of Triangular and Tetrahedral Structures
  • 3.1 Game Playing
  • 3.2 Concentrating Pressure
  • 3.3 Natural Forms
  • 3.4 Stability
  • 3.5 Computer Displays
  
  
• 4 The Cost of Numeric Representations
  • 4.1 Integers
  • 4.2 Rational Numbers
  • 4.3 Irrational Numbers
  • 4.4 Floating Point
  
  
• 5 Construction of Coordinate Systems
  • 5.1 Polyhedra as Coordinate Systems
  • 5.2 Cubic Coordinates
  • 5.3 Introducing Tetrahedral Coordinates
  • 5.4 Natural Coordinates
  • 5.5 Tetrahedral Normalization
  • 5.6 Octahedral Coordinates
  • 5.7 RhombHex Coordinates
  • 5.8 Icosahedral Coordinates - Rough Sketch
  
  
• 6 Converting Coordinates
  • 6.1 Length of Base Vectors
  • 6.2 Preserving Units
  • 6.3 Natural to Cubic
  • 6.4 Cubic to Natural (Unnormalized)
  
  
• 7 Projecting Coordinates Into a Plane
  • 7.1 Square to Natural (Unnormalized)
  • 7.2 Natural to Square
  
  
• 8 Linear Relationships
  • 8.1 Cubic Coordinates Linear Equations
  • 8.2 Natural Coordinates Linear Equations
  
  
• 9 Distance
  • 9.1 Cubic Coordinates Distance
  • 9.2 Natural Coordinates (Unnormalized) Distance
  • 9.3 Octahedral Coordinates Distance
  • 9.4 RhombHex Coordinates Distance
  • 9.5 Rational Vector Lengths
  
  
• 10 Scaling
• 11 Translation
• 12 Rotation
  • 12.1 Cubic Rotation Functions
  • 12.2 Natural Rotation Functions
  • 12.3 Natural Rational Orientations
  
  
• 13 Geometric Products
  • 13.1 Scalar Product
  • 13.2 Vector Product
  • 13.3 Volume
  
  
• Bibliography

1 Introduction

Natural doesn't necessarily mean obvious, as Natural Logarithms may seem less obvious or simple than Common Logarithms, and three-phase electrical current less obvious than two-phase or direct current. Non-orthogonal coordinate bases may appear less intuitive than orthogonal but seem more natural, primitive and balanced because nature seems to favor its orientations over the orthogonality favored by Pythagoras and his followers. It may not be possible to strip away every Pythagorean-based device and see the alternative where sixths of a circle are preferred instead of fourths in the foundations of our coordinates, but there seems to be value to Natural Coordinates as an alternative way of looking at things.

2 Background and Acknowledgements

The process of producing the Natural Coordinate formulae of this paper was manual and error-prone, starting with minimum equidistant axes positioned relative to Cubic Coordinate axes, formulating conversions, and drawing formulae through the conversions while tinkering with the results to simplify and rework traditional orthogonally-biased constructs.

Natural Coordinate formulae presented in following sections were independently discovered by this author, and still represent the author's preferred form. Along the way, Fuller's Synergetics[1] was discovered at the local library. This was later discovered on the Internet together with Darrel Jarmusch[2] in a Usenet posting on a Tetrahedral distance formula, Urner's pages about Quadrays[3], and, most-recently, Tom Ace[4]. No doubt others have followed similar paths.

3 Uses of Triangular and Tetrahedral Structures

Many cases have surfaced in this author's experience where non-orthogonal representation is an advantage. Here are a few applications of triangles and tetrahedra.

3.1 Game Playing

The fields of game boards are frequently arranged hexagonally instead of orthogonally. This results from an attempt to accomplish better packing of locations on the board. The hexagons are simply optimally-packed circles with the edges flattened a bit. This is one layer of the optimal sphere packing that Fuller[1] shows is optimized under a Natural Coordinate system. Join the centers and you have triangles - triangle and hex grids are quite related. Should grids of real cities be laid out triangularly, too?

As a boy, this author created n-dimensional variations of checkers and chess. A system based upon n-dimensional triangulation produced a better field to play in than a rectangular field. To add a player in a triangular system, add a dimension, and the new player and movements are equally related to all other players. How to add a player or dimension in an orthogonal system is less obvious, more complex, and does not result in symmetry since any edge of a rectangle is related differently to the opposite versus adjacent edges - a complication which is raised to the nth degree as you add dimensions. That was when this author first started thinking about Tetrahedral representations of space.

3.2 Concentrating Pressure

Early machines to produce artificial diamonds used hydraulic pistons in a tetrahedral arrangement to concentrate all their force on a single point at the center of the tetrahedron. This is a form that is often repeated when applying forces to a point.

3.3 Natural Forms

Crystals often form the naturally-stable tetrahedral shapes that occur when spheres are packed together.

3.4 Stability

An architect's work often involves reinforcing inherently-unstable rectangles with triangular cross members to make them stable. Stability is generally created using triangles, even if the external form is designed to look like rectangles. One might ask whether the external form should be rectangular to start with. Fuller's geodesic dome shows what can be accomplished using triangles symmetrically at one extreme, but it is not unthinkable that architecture in general should use triangles, tetrahedra, octahedra, and other forms more directly instead of hiding their use as supports for unstable rectangles.

3.5 Computer Displays

The appearance of a computer screen might be enhanced if the pixels were differently packed, which might make it possible to represent natural objects better. Display algorithms might be improved to take advantage of this in dithering and non-orthogonal rotation of bitmaps. The impact of more-natural packing could extend beyond conventional displays to new media and storage devices. While this might require significant redesign of models and interfaces, it could be worth investigating.

4 The Cost of Numeric Representations

The choice of coordinate axes in a geometric model affects whether numbers in representations of specific objects are rational or irrational, which in turn may dictate whether resulting computed values are exact or inexact. Where they have a choice, users of a coordinate system may tend to use those objects and transformations that are easier to represent.

4.1 Integers

Integers, especially those of limited range, are easiest to store exactly. They also are perceived as easier to use. Unsigned integers may be enough for limited regions or using Tetrahedral Coordinates (see section 5.3), simplifying the representation further.

4.2 Rational Numbers

The advantage of rational numbers is that they relieve the model designer from the task of choosing a base unit which is small enough to adequately represent all resulting objects in the system.

Rationals may be exactly represented by an integer numerator and an integer denominator with common factors divided out, but the maximum size of exact rational representations resulting from arbitrary sets of transformations is not fixed even within a limited coordinate range.

Rationals may be inexactly represented using floating point, accumulating unbound error instead of unbound size.

4.3 Irrational Numbers

Irrational numbers are required to represent objects not naturally aligned with the coordinate system. Certain irrational numbers are found in this paper used to preserve units in conversions between non-aligned coordinate systems, and converting between radian angle values and pizza-like division of a circle into equal arcs. For the purposes of this paper, objects which align without irrational relationships in a model will be considered to be more natural.

Specific classes of irrational numbers could be represented exactly and later rationalized to whatever accuracy is required so that a properly-engineered pocket calculator would not display results flawed by internal truncation, even without understanding the error characteristics and tolerance of the problem being solved. But this adds complexity problems to size problems - like those described above for rationals - in extended chains of formulations with exact irrationals.

The desire to avoid irrationals significantly biases most users, for example, of a rectangular coordinate system towards rectangular objects and views of those objects, even if this is not the best form of an object or view for a particular purpose by other criteria. Such biases make coordinate systems not all equal in representing specific types of objects.

4.3.1 Continued Fractions

Irrational results can sometimes be exactly represented as series of terms of continued fractions[7]. There could be exploitable order in the series that permits useful generalization between irrationals of various classes.

4.4 Floating Point

Computer programs may immediately convert numbers into floating-point representations, which are only approximations in the case of irrationals and also most rationals. Manipulation of floating point can be a huge source of error caused by computer programs.

The advantage of floating point over exact representations is that it's storage is of a fixed size. A disadvantage is that the truncation of precision, though automatic, must be manually accounted at every atomic and composite operation, which often involves operating differently on different ranges of inputs and intermediate results and more. Otherwise truncation of precision produces a loss that is more devastating than unmanaged storage of exact results because erroneous results are less obvious than an overgrown exact representation.

A few years ago, reports of a bug in Pentium hardware caused people all over the world to report floating point errors that generally turned out to be software mismanagement of floating point that was working as designed but producing hugely flawed results in simple cases. Though the hardware be correct, the transistors are wasted if the representation is inadequate to the operations. Floating point may not be easily applied to generally solve apparently-simple computations, such as the quadratic formula.

5 Construction of Coordinate Systems

5.1 Polyhedra as Coordinate Systems

To generate a coordinate system, choose a polyhedron and include one base vector orthogonal to each face, producing quadrants at the vertices. The following are clearly not the only polyhedra that might generate interesting coordinate systems for three-dimensional (four-directional as Fuller[1] would have it) space as we know it.

Polyhedron	Faces/Bases	Coordinate characteristics	Vertices/Coordinate Zones
Cube	6	3 signed, orthogonal	8
Regular Tetrahedron	4	4 unsigned, natural	4
Regular Octahedron	8	4 signed, natural	6
Rhombic Hexahedron	6	3 signed, natural	8
Rhombic Dodecahedron	12	6 signed	14
Regular Dodecahedron	12	6 signed, golden ratio	20
Cuboctahedron	14	7 signed, natural and orthogonal	12
Regular Icosahedron	20	10 signed, natural and golden	12

5.2 Cubic Coordinates

Based upon the faces of a cube, Cubic Coordinates represent space via six orthogonal base vectors emanating from the origin where:

$$\begin{eqnarray*} X_{p} & = & \textrm{positive x axis}\\ X_{n} & = & \textrm{ne... ...\textrm{positive z axis}\\ Z_{n} & = & \textrm{negative z axis} \end{eqnarray*}$$

This is the age-old popular representation of space. A 6-tuple of unsigned coefficients $\left(x_{p},x_{n},y_{p},y_{n},z_{p},z_{n}\right)$ scales these base vectors in the sum:

$$x_{p}\cdot X_{p}+x_{n}\cdot X_{n}+y_{p}\cdot Y_{p}+y_{n}\cdot Y_{n}+z_{p}\cdot Z_{p}+z_{n}\cdot Z_{n}$$

This addresses all points in the corresponding space.

5.2.1 Signed Coordinates

Where a coordinate system has pairs of opposite bases such as the Cubic bases $X_{p}$ and $X_{n}$, a corresponding coefficient may be subtracted from the opposite coefficient and zeroed, replacing the pair with a single signed coordinate. The Cubic 6-tuple is thus compressed to a signed 3-tuple $\left(x,y,z\right)$ where:

$$\begin{eqnarray*} x & = & x_{p}-x_{n}\\ y & = & y_{p}-y_{n}\\ z & = & z_{p}-z_{n} \end{eqnarray*}$$

With the orthogonal bases of Cubic Coordinates, this produces a unique representation of a vector where the signs (or zeros) of the coefficients identify the zones containing a vector.

5.3 Introducing Tetrahedral Coordinates

Based upon the faces of a regular tetrahedron, Tetrahedral Coordinates are a Natural Coordinate system which represent space via four base vectors, A, B, C, and D, emanating from the origin. These base vectors point from the center to each corner of a regular tetrahedron.

The four spikes of a caltrop are also $\left(A,B,C,D\right)$, which also point from the center of a cube to 4 of eight alternating corners (see James Bond movie Tomorrow Never Dies, where the fancy BMW drops caltrops to flatten tires of following cars). Each vector is positioned symmetrically with respect to each other vector. By adding scaled versions of these vectors, any point in space may be reached. Hence, these coordinates may address any point by applying an appropriate 4-tuple of coefficients $\left(a,b,c,d\right)$ to the sum:

$$a\cdot A+b\cdot B+c\cdot C+d\cdot D$$

There are no pairs of opposites in this coordinate system as there are in the Cubic Coordinate system, so the coordinates are unsigned.

5.4 Natural Coordinates

The Tetrahedral Coordinate system is the first of several Natural Coordinate systems described here, which are based upon four coordinates, each of which is balanced by the sum of the other three, making it possible to add or subtract any number from all coordinates and reach the same position in space. For example, the following Natural 4-tuples all represent the same point in space:

$$\begin{array}{c} \left(\begin{array}{cccc} 7, & 0, & 0, & ... ...{array}{cccc} 0, & -7, & -7, & -7\end{array}\right)\end{array}$$

5.4.1 Natural Normalizations

Natural coordinate formulae may produce multiple representations of vectors. A normalization function $norm\left(a,b,c,d\right)$ may be used to adjust each coordinate of a newly-produced natural vector by the same amount as follows:

$$\begin{eqnarray*} a_{n} & = & a+norm\left(a,b,c,d\right)\\ b_{n} & = & b+norm\l... ...rm\left(a,b,c,d\right)\\ d_{n} & = & d+norm\left(a,b,c,d\right) \end{eqnarray*}$$

To produce unique representations of coordinates, the normalization function must normalize all equivalent vectors to the same representation, satisfying for arbitrary values of $k$ the condition:

$$\begin{array}{ccc} norm\left(a,b,c,d\right) & = & norm\left(a+k,b+k,c+k,d+k\right)+k\end{array}$$

The distinguishing characteristic of Natural Coordinate systems presented in other sections is the choice of a suitable function $norm\left(a,b,c,d\right)$ to normalize the coordinates. The normalization may also be used to abbreviate the number of coordinates in a representation to three. Formulae may be simplified to assume a particular normalization type and automatically produce results normalized in that type. To convert to a different Natural Coordinate system, re-normalize the four coordinates using the target Natural Coordinate system's normalization function.

5.5 Tetrahedral Normalization

To normalize non-negative Tetrahedral Coordinates, you can subtract the least coordinate value from the others, performing the normalization:

$$\begin{eqnarray*} norm_{t}\left(a,b,c,d\right) & = & -\min \left(a,b,c,d\right) \end{eqnarray*}$$

This choice of normalization function produces minimal non-negative coordinate values, which is essential since there are no negative Tetrahedral bases. The coordinate of the axes opposite the (tetrahedrally-shaped) coordinate zones that contain the vector becomes zero. At least one coordinate will be zero since a vector always exists in at least one zone. A vector enters multiple zones only where it lies exactly on the boundary between them.

Tetrahedral normalization seems to be the least-practical Natural normalization of the three presented here. Since the zero is not introduced at a constant position, trying to abbreviate Tetrahedrally-normalized vectors to three coordinates is cumbersome at best, also making it difficult to simplify or automatically normalize formulae. For this reason, no Tetrahedrally-optimized formulae are presented here, but manual Tetrahedral normalization can be applied to the results of any of the general Natural formulae that follow.

If vectors were confined to a single zone - one fourth of space - then the zero is introduced at a constant location, but this form is handled already by RhombHex Coordinates as described in following sections, which can be unsigned as long as vectors remain in the proper zone.

5.6 Octahedral Coordinates

Based upon the faces of a regular octahedron, Octahedral Coordinates are Natural Coordinates, like Tetrahedral Coordinates except that there are negative base vectors opposing each positive one so the 4-tuple of coordinates is signed instead of unsigned (see section 5.2.1).

5.6.1 Octahedral Normalization

As Tom Ace[4] shows, signed Natural Coordinate 4-tuples should be normalized to a zero sum, as follows:

$$\begin{eqnarray*} norm_{o}\left(a,b,c,d\right) & = & -\frac{a+b+c+d}{4} \end{eqnarray*}$$

This choice of normalization simplifies distance and related formulae. This normalization produces distinct vector coordinates in which three of the four coordinates are never identical.

5.6.2 Octahedral Abbreviation

Normalized Octahedral 4-tuples may be abbreviated as 3-tuples, since any coordinate may be reconstructed from the other three as the negative of their sum, but they continue to be represented as a 4-tuple in this paper. To optimize the Octahedral formulae presented in following sections for abbreviation, substitute $-\left(a+b+c\right)$ for $d$.

5.7 RhombHex Coordinates

RhombHex Coordinates are Natural Coordinates where normalization always zeros the same axis, yielding a coordinate system with six base vectors and three coordinates. With the elimination of one axis, two sides of the regular octahedron have been lost, leaving a hexahedron with rhomb sides to generate this coordinate system.

5.7.1 RhombHex Normalization

The signed 4-tuple is compressed to a signed 3-tuple by inserting a zero at a known location, normalizing as follows:

$$\begin{eqnarray*} norm_{r}\left(a,b,c,d\right) & = & -d \end{eqnarray*}$$

RhombHex Coordinates are represented in following sections as a 3-tuple instead of a 4-tuple, because $d$ is always zero. In this form, the signs (or zeros) of the remaining coordinates indicate which RhombHex Coordinate zone the vector enters.

5.8 Icosahedral Coordinates - Rough Sketch

This goes well beyond the intended scope of this paper, but each of the 10 icosahedral coordinate axes participate in two distinct Natural Coordinate subsystems as well as other asymmetrical coordinate subsystems.

5.8.1 Icosahedral Normalization

At first glance, it is possible to adjust any coordinate of a point by choosing either of the two Natural Coordinate subsystems it participates in. Also, for coordinates containing powers of the golden ratio, these can clearly be factored out using asymmetrical coordinate subsystems that permit $\sqrt{5}$ on one axis to be substituted by integers on others. The compound normalization possibilities make possible a rich set of natural primitive objects.

6 Converting Coordinates

There are an infinite number of ways to position one coordinate system inside of another. But for symmetry, it seems best to place the Tetrahedral bases as previously described emanating from the center of a cube (at the origin of Cubic Coordinates) to four of the eight alternating corners. A more specific convenient construction follows.

6.1 Length of Base Vectors

The length of the base vector of each coordinate system will determine whether specific polyhedra will have rational edge lengths.

6.1.1 Cubic Base Length

The length of a base vector in cubic coordinates has been established at 1, since this establishes the length of the edges of the smallest cube with vertices at integer coordinates at 1.

6.1.2 Natural Base Length

As Urner pointed out, the length of a base vector in a Natural coordinate system need not be 1. Using a base vector length of $\sqrt{\frac{3}{8}}$ establishes the length of the edges of the smallest regular tetrahedron with vertices at integer coordinates at 1.

6.2 Preserving Units

When converting between non-aligned coordinate systems, irrational numbers must be used to preserve the lengths of vectors between coordinate systems. Most computing systems handle irrational numbers inexactly. The alternative is to permit vectors as represented in the different coordinate systems to have different proportional lengths. Since irrational numbers are not completely computable anyway by rational mechanisms, the irrational scaling factor will likely be substituted at some point with a convenient rational number which will not perfectly preserve scale of units.

6.3 Natural to Cubic

6.3.1 Cubic Base Vectors

Symmetrically construct Cubic base vectors from Tetrahedral base vectors as follows:

$$\begin{eqnarray*} X_{p} & = & \sqrt{2}\cdot \left(A+B\right)\\ X_{n} & = & \sqr... ...t \left(A+D\right)\\ Z_{n} & = & \sqrt{2}\cdot \left(B+C\right) \end{eqnarray*}$$

6.3.2 Conversion to Cubic

To convert an arbitrary vector from Natural Coordinates to Cubic Coordinates, compute as follows:

$$\begin{eqnarray*} x & = & \sqrt{\frac{1}{8}}\cdot \left(a+b-c-d\right)\\ y & = ... ...d\right)\\ z & = & \sqrt{\frac{1}{8}}\cdot \left(a-b-c+d\right) \end{eqnarray*}$$

6.4 Cubic to Natural (Unnormalized)

6.4.1 Natural Base Vectors

Symmetrically construct Tetrahedral base vectors from Cubic base vectors as follows:

$$\begin{eqnarray*} A & = & \sqrt{\frac{1}{8}}\cdot \left(X_{p}+Y_{p}+Z_{p}\right)... ... D & = & \sqrt{\frac{1}{8}}\cdot \left(X_{n}+Y_{n}+Z_{p}\right) \end{eqnarray*}$$

6.4.2 Conversion to Natural (Unnormalized)

To convert an arbitrary vector from Cubic Coordinates to Natural Coordinates, compute as follows:

$$\begin{eqnarray*} a & = & \sqrt{\frac{1}{2}}\cdot \left(x+y+z\right)\\ b & = & ... ...-z\right)\\ d & = & \sqrt{\frac{1}{2}}\cdot \left(-x-y+z\right) \end{eqnarray*}$$

Then apply the appropriate normalization to the result.

7 Projecting Coordinates Into a Plane

It is common to abbreviate graphs in Cubic Coordinates to the surface of a plane by omitting the coordinate of one dimension orthogonal to the plane. A similar approach may be taken in Natural Coordinate systems, eliminating the coordinate of one base vector that is orthogonal to the plane and projecting the remaining base vectors into the plane, which preserves their natural balance. Using a base vector length of 1 establishes the length of the edges of the smallest regular triangle with vertices at integer coordinates at 1, scaling the lengths of vectors parallel to the projected plane by $\sqrt{3}$ unless other adjustment is applied. The vector may first be scaled by the ratio between the length of a base vector of the truncated axis and the scalar product (see section 13.1) of the base vector with the vector to produce a perspective projection instead of an orthogonal projection.

The results of projecting an object of a Natural Coordinate system into a plane (using a Natural axis) or converting to Cubic Coordinates and projecting into a plane will be different because the axes of Cubic and Natural space were not aligned. Other conversions are possible between Natural and Cubic Coordinates are possible which keep a single axis aligned, but it is not possible for a conversion to simultaneously align all axes. Or, perhaps more practically, it is possible to define conversions directly between coordinate planes after the fact of some projection using the coordinate sets $\left(a,b,c\right)$ and $\left(x,y\right)$ respectively. It is easiest to choose one axis to be aligned in the resulting planar mapping. In the following example, we choose to align the positive x axis with the positive a axis. Other alignments may be accomplished in a similar fashion.

7.1 Square to Natural (Unnormalized)

Omitting a redefinition of the axes, to convert an arbitrary vector from square coordinates to Natural Coordinates keeping the x axis aligned with the a axis, compute as follows:

$$\begin{eqnarray*} a & = & x\\ b & = & \sqrt{\frac{1}{3}}\cdot y\\ c & = & -\sqrt{\frac{1}{3}}\cdot y \end{eqnarray*}$$

Then apply the appropriate normalization.

As in the higher-order Natural Coordinate systems, you can normalize the by adding or subtracting constants equally to all coordinates. The triangular coordinate system is the natural one with only positive coordinates, which may be normalized to make one coordinate zero. If negative coordinates are permitted, the coordinate system may be hexagonal, which may use normalization similar to what was described for the higher-order equivalents, etc.

The following sections describing linear relationships, distance, rotation, and scalar products using Natural Coordinates may be easily adapted to the lower-order Natural Coordinate systems, for example, the distance function for length of a vector becomes:

$$h^{2}=\frac{3}{2}\cdot \left(a^{2}+b^{2}+c^{2}\right)-\frac{1}{2}\cdot \left(a+b+c\right)^{2}$$

7.2 Natural to Square

Omitting a redefinition of the axes, to convert an arbitrary vector from Natural Coordinates to square coordinates keeping the a axis aligned with the x axis, compute as follows:

$$\begin{eqnarray*} x & = & a-\frac{1}{2}\cdot \left(b+c\right)\\ y & = & \sqrt{\frac{3}{4}}\cdot \left(b-c\right) \end{eqnarray*}$$

The following sections describing linear relationships, distance, rotation, and scalar products using Cubic Coordinates may be easily adapted to the lower-order square coordinate system.

8 Linear Relationships

The set of points in a space with a specific linear relationship between the Cubic or Natural Coordinates form a plane. Taking all points on one side of some such plane divides space in half producing a half-space. Where there is a relationship of linear equality describing the points on the surface of a plane, an inequality such as greater-than or less-than describes a half-space. By joining and intersecting different half-spaces, it is possible to describe flat-sided solid objects, of which the tetrahedron is the simplest finite form with a non-empty volume.

8.1 Cubic Coordinates Linear Equations

In Cubic Coordinates, a plane may be represented by the 4-tuple of coefficients $\left(E,F,G,H\right)$ of the equation:

$$E\cdot x+F\cdot y+G\cdot z=H$$

This plane is orthogonal to the Cubic Coordinates' vector $\left(E,F,G\right)$. By partitioning the set of all planes into two sets, those containing the origin, and those not containing the origin, the required number of coordinates could be reduced to 3 with an extra mark to indicate which set the plane belongs to. Planes not going through the origin can rely on H being normalized to $1$ or such that $\left(E,F,G\right)$ represents the vector from the origin to the plane, orthogonal to the plane. Planes going through the origin can be represented by any orthogonal vector.

8.2 Natural Coordinates Linear Equations

In Natural Coordinates, a plane may be represented by the 5-tuple of coefficients $\left(M,N,P,Q,H\right)$ of the equation:

$$M\cdot a+N\cdot b+P\cdot c+Q\cdot d=H$$

This plane is orthogonal to the Natural Coordinates' vector $\left(M,N,P,Q\right)$. By partitioning the set of all planes or reverting to signed coefficients, the number of coefficients could be further reduced as described for Cubic planes (section 8.1). Since there is a direct correspondence between the plane and the orthogonal vector, normalization can be used to abbreviate the representation (section 5.4.1).

9 Distance

Distance formulae permit computation of the length of a vector not aligned with the axes of the coordinate system, but rotated in some other orientation. This also makes it possible to describe shapes such as spheres and other ellipsoids by the distance from the foci to the surface.

9.1 Cubic Coordinates Distance

The length $h$ of a vector is related to the signed 3-tuple of Cubic Coordinates $\left(x,y,z\right)$ by the formula:

$$h^{2}=x^{2}+y^{2}+z^{2}$$

Since opposite vectors are collapsed and squaring signed numbers produces unsigned numbers, only three terms are required.

9.2 Natural Coordinates (Unnormalized) Distance

The length $h$ of a vector is related to the signed 4-tuple $\left(a,b,c,d\right)$ by the formula:

$$h^{2}=\frac{1}{2}\cdot \left(a^{2}+b^{2}+c^{2}+d^{2}\right)-\frac{1}{8}\cdot \left(a+b+c+d\right)^{2}$$

9.3 Octahedral Coordinates Distance

Simplifying the Natural Coordinates distance formula for Octahedral normalization leaves:

$$h^{2}=\frac{1}{2}\cdot \left(a^{2}+b^{2}+c^{2}+d^{2}\right)$$

9.4 RhombHex Coordinates Distance

Simplifying the Natural Coordinates distance formula for RhombHex normalization leaves:

$$h^{2}=\frac{1}{2}\cdot \left(a^{2}+b^{2}+c^{2}\right)-\frac{1}{8}\cdot \left(a+b+c\right)^{2}$$

9.5 Rational Vector Lengths

The length of a vector is the distance from the origin to the position identified by the coordinates.

While the set of vectors with rational lengths is closed under operations such as scaling, rotation, and translation (using rational factors), it is not closed under common operations such addition, subtraction, or vector products of non-parallel vectors. If vectors with rational lengths should represent natural objects then when deriving a vector using these operations it is necessary to adjust it to rationalize its length to make it represent something natural. When constructing natural objects out of packed particles of uniform size, this type of adjustment would clearly be necessary.

10 Scaling

Cubic and Natural vectors may all be scaled by multiplying each coordinate by the same scalar value.

11 Translation

Cubic vectors may be added for translation by adding corresponding coordinates. Natural coordinate vectors may likewise be added for translation by adding corresponding coordinates. When adding two Octahedral coordinate vectors or two RhombHex coordinate vectors, the normalization is preserved. When adding two Tetrahedral vectors or two Natural vectors of different types, re-normalization will be required.

12 Rotation

Rotation is a vector transformation which may change the orientation of the vector but never the length of the vector or of sums, differences, or products of sets of like-rotated vectors. Matrix multiplication is a good way to accomplish rotation. Multiple rotations may be concatenated into a single matrix by multiplying the matrices. When a vector is transformed using a rational matrix, if the coordinates were all rational, then the coordinates of the resulting vector are also rational. This is true not only of explicitly transformed vectors but also of sums, differences, or products of like-transformed vectors.

12.1 Cubic Rotation Functions

Cosine and sine identify the $X_{p}$ and $Y_{p}$ components of $X_{p}'$ (and also of $Y_{p}'$ by transposition) as rotated around $Z_{p}$. Any pair of numbers $(\cos ,\sin )$ where:

$$\begin{eqnarray*} \cos & : & -1\leq \cos \leq 1\\ \sin & : & -1\leq \sin \leq 1 \end{eqnarray*}$$

(such that)

$$\begin{eqnarray*} \cos ^{2}+\sin ^{2} & = & 1 \end{eqnarray*}$$

These represent a Cubic rotation.

12.1.1 Rational Cubic Rotation

For radian angle $\theta$ : $-\pi <\theta <\pi$, consistent rational values for sine and cosine may be found from rational value $r\approx \tan \left(\frac{\theta }{2}\right)$, which is easily obtained from a continued fraction[7], as follows:

$$\begin{eqnarray*} \cos \left(r\right) & = & \frac{1-r^{2}}{r^{2}+1}\\ \sin \left(r\right) & = & \frac{2\cdot r}{r^{2}+1} \end{eqnarray*}$$

This produces Cubic rotation that is rational and preserves rational vector lengths[8]. Simple range reduction techniques avoid $\theta \approx \pm \pi$ and complete the circle. Except where r is $\pm 1$ ($\theta$ is $\pm \frac{\pi }{2}$), Cubic rational rotation never evenly divides a circle, so the combination of arbitrary or repeated rational rotations generally has a less-simple representation than the original rotations. Irrational rotations are required for other even divisions of the circle in Cubic Coordinates.

12.1.2 Cubic Rotation by Matrix

The rotation functions are easily applied as a transformation in a matrix as follows:

$$\left[\begin{array}{ccc} \cos & \sin & 0\\ -\sin & \cos & 0\\ 0 & 0 & 1\end{array}\right]$$

Rotations around other axes may be derived by migrating sines and cosines down the main diagonal of an identity matrix. Cubic rotations may be aggregated by matrix multiplication.

12.2 Natural Rotation Functions

New functions which this author designates aota, bota, and cota identify the $A$, $B$ and $C$ components of $A'$ (and also of $B'$ and $C'$ by transposition) as rotated around $D$. Any 3-tuple of numbers $\left(aota,bota,cota\right)$ where:

$$\begin{eqnarray*} aota & : & -\frac{2}{3}\leq aota\leq \frac{2}{3}\\ bota & : &... ... \frac{2}{3}\\ cota & : & -\frac{2}{3}\leq cota\leq \frac{2}{3} \end{eqnarray*}$$

(such that)

$$\begin{array}{ccc} aota^{2}+aota\cdot bota+bota^{2} & = & \... ...ota^{2} & = & \frac{1}{3}\\ aota+bota+cota & = & 0\end{array}$$

These represent a Natural rotation.

12.2.1 Abstract Angle Functions for Natural Rotation

The Natural rotation component functions of radian angle $\theta$ may be derived from the similar cosine function as follows:

$$\begin{eqnarray*} aota\left(\theta \right) & = & \frac{2}{3}\cos \left(\theta \r... ...& = & \frac{2}{3}\cos \left(\theta +\frac{4\cdot \pi }{3}\right) \end{eqnarray*}$$

These functions are $\frac{2\pi }{3}$ out of phase with each other, rather than $\frac{\pi }{2}$ as sine and cosine, and are scaled by $\frac{2}{3}$ with respect to sine and cosine.

12.2.2 Rational Natural Rotation

For radian angle $\theta$ : $-\pi <\theta <\pi$, consistent rational values for aota, bota, and cota may found from rational value $r\approx \sqrt{3}\cdot \tan \left(\frac{\theta }{2}\right)$ as follows:

$$\begin{eqnarray*} aota\left(r\right) & = & \frac{6-2\cdot r^{2}}{3\cdot r^{2}+9}... ...cota\left(r\right) & = & \frac{r^{2}+6\cdot r-3}{3\cdot r^{2}+9} \end{eqnarray*}$$

This produces Natural rotation that is rational and preserves rational vector lengths[9]. Simple range reduction techniques avoid $\theta \approx \pm \pi$ and complete the circle. Except where r is $\pm 1$ (in radians $\theta$ is $\pm \frac{\pi }{3}$), Natural rational rotations never evenly divide a circle, so the combination of arbitrary or repeated rational rotations generally has a less-simple representation than the original rotations. Irrational rotations are required for other even divisions of the circle in Natural Coordinates.

12.2.3 Natural Rotation by Matrix

The rotation functions are easily applied as a transformation matrix as follows:

$$\left[\begin{array}{cccc} aota & bota & cota & -\frac{1}{3}... ... cota & aota & -\frac{1}{3}\\ 0 & 0 & 0 & 1\end{array}\right]$$

The results of this transformation are not normalized. Natural rotations around other base vectors may be constructed by appropriately migrating the values in the matrix. Natural rotations may be aggregated by matrix multiplication.

12.2.4 Octahedral Rotation by Matrix

Optimizing the Natural rotation matrix for Octahedral normalization yields:

$$\left[\begin{array}{cccc} aota & bota & cota & -\frac{1}{3}... ...} & -\frac{1}{3} & -\frac{1}{3} & \frac{2}{3}\end{array}\right]$$

This produces normalized results. It is unnecessary to compute all four coordinates this way, since after computing any three, the fourth is the negative of their sum. Octahedral rotations around other base vectors may be constructed by appropriately migrating the values in the matrix. Octahedral rotations may be aggregated by matrix multiplication.

12.2.5 RhombHex Rotation by Matrix

Optimizing the Natural rotation matrix for RhombHex normalization yields:

$$\left[\begin{array}{ccc} aota+\frac{1}{3} & bota+\frac{1}{3... ...c{1}{3} & cota+\frac{1}{3} & aota+\frac{1}{3}\end{array}\right]$$

This produces normalized results. RhombHex rotational matrices around other axes are not symmetrical so they cannot be produced by migrating this RhombHex matrix down the main diagonal. To produce RhombHex rotations around other axes, start by appropriately migrating the Natural rotation matrix, RhombHex-normalize the rows, and then drop the fourth row and column. RhombHex rotations may be aggregated by matrix multiplication.

12.3 Natural Rational Orientations

Either system supports rational orientations and irrational orientations. Rational orientations are the result of being reachable by a rational rotations, as previously described. There is clearly a difference in which orientations can be reached rationally. Each system seems to permit a single basic rational rotation which divides a circle into even parts such that it can be repeated without an unbounded increase in complexity of the terms (or resorting to inexact numbers). So Natural Coordinates effectively prefer triangular rotations to orthogonal rotations. If triangular orientations are more prevalent in nature than orthogonal rotations, then Natural Coordinate systems seem more Natural than the Cubic Coordinate system.

12.3.1 The $\frac{\pi }{3}$ or Triangular Rotation

$$\begin{eqnarray*} \cos & = & \frac{1}{2}\\ \sin & = & \frac{\sqrt{3}}{2}\\ aot... ...\frac{1}{3}\\ bota & = & -\frac{2}{3}\\ cota & = & \frac{1}{3} \end{eqnarray*}$$

12.3.2 The $\frac{\pi }{2}$ or Orthogonal Rotation

$$\begin{eqnarray*} \cos & = & 0\\ \sin & = & 1\\ aota & = & 0\\ bota & = & -\frac{\sqrt{3}}{3}\\ cota & = & \frac{\sqrt{3}}{3} \end{eqnarray*}$$

13 Geometric Products

Sir William Rowan Hamilton[4] wrote about scalar and vector products, which were developed into Geometric Algebra by Grassmann[5] and Clifford[6].

13.1 Scalar Product

The scalar product of two vectors is a scalar equal to the product of the lengths as one is projected orthogonally into the other. The scalar product has been compared with the cosine function in that it identifies the component after angular displacement that is parallel to the original vector. Although the role of the cosine in Natural Coordinate systems has been replaced by a new function aota, there is no angular difference between aota and cosine that would imply an alternative scalar product for Natural systems.

13.1.1 Cubic Coordinates Scalar Product

Where:

$$\begin{eqnarray*} V_{1} & = & \left(x_{1},y_{1},z_{1}\right)\\ V_{2} & = & \left(x_{2},y_{2},z_{2}\right) \end{eqnarray*}$$

The scalar product $S=V_{1}\cdot V_{2}$ may be computed:

$$S_{12}=x_{1}\cdot x_{2}+y_{1}\cdot y_{2}+z_{1}\cdot z_{2}$$

13.1.2 Natural Coordinates (Unnormalized) Scalar Product

Where:

$$\begin{eqnarray*} V_{1} & = & \left(a_{1},b_{1},c_{1},d_{1}\right)\\ V_{2} & = & \left(a_{2},b_{2},c_{2},d_{2}\right) \end{eqnarray*}$$

The scalar product $S=V_{1}\cdot V_{2}$ may be computed:

$$S_{12}=\frac{1}{2}\cdot \left(a_{1}\cdot a_{2}+b_{1}\cdot b_... ...1}+c_{1}+d_{1}\right)\cdot \left(a_{2}+b_{2}+c_{2}+d_{2}\right)$$

13.1.3 Octahedral Coordinates Scalar Product

Simplifying the Natural Coordinates scalar product formula for Octahedral normalization leaves:

$$S_{12}=\frac{1}{2}\cdot \left(a_{1}\cdot a_{2}+b_{1}\cdot b_{2}+c_{1}\cdot c_{2}+d_{1}\cdot d_{2}\right)$$

Note that this formula is more similar to the RhombHex form if using Octahedral abbreviation (section 5.6.2).

13.1.4 RhombHex Coordinates Scalar Product

Simplifying the Natural Coordinates scalar product formula for RhombHex normalization leaves:

$$S_{12}=\frac{1}{2}\cdot \left(a_{1}\cdot a_{2}+b_{1}\cdot b_... ...ft(a_{1}+b_{1}+c_{1}\right)\cdot \left(a_{2}+b_{2}+c_{2}\right)$$

13.2 Vector Product

The product of two vectors is a bivector representing a directed area of the displaced angle. The vector product is like a sine, yielding the area after angular displacement orthogonal to the original vectors. It might be practical to replace the bivector with multiple non-orthogonal vector products more-resembling bota and cota, which might be conventional vectors rather than bivectors because there are two, but this section as presently written simply converts the conventional product to get at volume in the next section.

13.2.1 Cubic Coordinates Vector Product

Where:

$$\begin{eqnarray*} V_{1} & = & \left(x_{1},y_{1},z_{1}\right)\\ V_{2} & = & \lef... ...},z_{2}\right)\\ V_{12} & = & \left(x_{12},y_{12},z_{12}\right) \end{eqnarray*}$$

The product $V_{12}=V_{1}\times V_{2}$ may be found as the coefficients of base vectors $X$, $Y$, $Z$ produced by the pseudo-determinant:

$$\left\vert\begin{array}{ccc} X & Y & Z\\ x_{1} & y_{1} & z_{1}\\ x_{2} & y_{2} & z_{2}\end{array}\right\vert$$

Note that this is not a real determinant since it contains vectors, but it easily produces the real determinants of the coordinates of the vector product.

13.2.2 Natural Coordinates (Unnormalized) Vector Product

Where:

$$\begin{eqnarray*} V_{1} & = & \left(a_{1},b_{1},c_{1},d_{1}\right)\\ V_{2} & = ... ...\right)\\ V_{12} & = & \left(a_{12},b_{12},c_{12},d_{12}\right) \end{eqnarray*}$$

The product $V_{12}=V_{1}\times V_{2}$ may be found as the coefficients of base vectors $A$, $B$, $C$, $D$ produced by the pseudo-determinant:

$$\left\vert\begin{array}{cccc} A & B & C & D\\ 1 & 1 & 1 &... ...& d_{1}\\ a_{2} & b_{2} & c_{2} & d_{2}\end{array}\right\vert$$

This is not a real determinant since it contains vectors, but it easily produces the real determinants of the coordinates of the vector product. The length of this vector product is scaled by $\sqrt{8}$ compared with the cubic vector product. Various forms may be produced by manipulation, for example the following form leads to simpler Octahedral and RhombHex forms later:

$$\begin{array}{ccc} a_{12} & = & 2\cdot \left(b_{1}\cdot c_{... ...{2}\right)+d_{2}\cdot \left(a_{1}+b_{1}+c_{1}\right)\end{array}$$

13.2.3 Octahedral Coordinates Vector Product

The above pseudo-determinant produces results normalized as Octahedral Coordinates. However these computations may be simplified if the inputs are also known to be normalized, for example:

$$\begin{array}{ccc} a_{12} & = & 2\cdot \left(b_{1}\cdot c_{... ...\cdot a_{2}+a_{1}\cdot c_{2}-d_{1}\cdot d_{2}\right)\end{array}$$

Note that it is not necessary to compute all four coordinates this way since any coordinate is the negative of the sum of the other three computed in normalized form.

13.2.4 RhombHex Coordinates Vector Product

Simplifying the Natural Coordinates formulae for RhombHex normalization leaves:

$$\begin{array}{ccc} a_{12} & = & \left\vert\begin{array}{cc}... ...a_{1}\\ c_{2} & a_{2}\end{array}\right\vert\right)\end{array}$$

This produces RhombHex normalized results from simple sums of three small determinants.

13.3 Volume

The scalar and vector products permit computation of the volume of arbitrary tetrahedra. Take the vector differences between any three vertices of a tetrahedron and the remaining vertex. The vector product of two of these vectors forms a vector orthogonal to the face of the tetrahedron which shares those two vectors, the length of which is twice the area of that face. The scalar product of that vector product with the third vector - the triple product - is six times the volume of the tetrahedron, since the volume of a pyramid is equal to one third of the base area times times the orthogonal height. Thus, for a tetrahedron with vertices R, S, T, and U, the following edge vectors could be chosen as follows:

$$\begin{eqnarray*} E_{1} & = & R-U\\ E_{2} & = & S-U\\ E_{3} & = & T-U \end{eqnarray*}$$

Volume formulae are conveniently represented as determinants.

13.3.1 Cubic Coordinates Volume

Where:

$$\begin{eqnarray*} E_{1} & = & \left(x_{1},y_{1},z_{1}\right)\\ E_{2} & = & \lef... ...y_{2},z_{2}\right)\\ E_{3} & = & \left(x_{3},y_{3},z_{3}\right) \end{eqnarray*}$$

The volume V of the tetrahedron may be computed in units of equilateral unit cubes as a determinant:

$$V=\frac{1}{6}\cdot \left\vert\begin{array}{ccc} x_{1} & y_{... ...& y_{2} & z_{2}\\ x_{3} & y_{3} & z_{3}\end{array}\right\vert$$

13.3.2 Cubic Unit of Volume

This volume formula establishes the unit of volume as the smallest non-empty cube by volume with vertices occupying integer coordinates.

13.3.3 Natural Coordinates (Unnormalized) Volume

Where:

$$\begin{eqnarray*} E_{1} & = & \left(a_{1},b_{1},c_{1},d_{1}\right)\\ E_{2} & = ... ...d_{2}\right)\\ E_{3} & = & \left(a_{3},b_{3},c_{3},d_{3}\right) \end{eqnarray*}$$

The volume V of the tetrahedron may be computed in units described in a following section as a determinant:

$$V=\left\vert\begin{array}{cccc} 1 & 1 & 1 & 1\\ a_{1} & b... ...& d_{2}\\ a_{3} & b_{3} & c_{3} & d_{3}\end{array}\right\vert$$

13.3.4 Octahedral Coordinates Volume

Octahedral normalization permits any three columns of the Natural volume determinant to be added to the remaining row to reduce the determinant size, for example:

$$V=\left\vert\begin{array}{cccc} 1 & 1 & 1 & 4\\ a_{1} & b... ...& b_{2} & c_{2}\\ a_{3} & b_{3} & c_{3}\end{array}\right\vert$$

This form already supports Octahedral abbreviation (section 5.6.2) and is computationally simpler than the symmetrical form that appeared here previously, which was derived from the Octahedral normalization scalar and vector product but was discarded in favor of the suggestion to use determinants[10].

13.3.5 RhombHex Coordinates Volume

Simplifying the Natural Coordinates volume formula for RhombHex normalization due to zeros in the last coordinate leaves:

$$V=\left\vert\begin{array}{ccc} a_{1} & b_{1} & c_{1}\\ a_... ...& b_{2} & c_{2}\\ a_{3} & b_{3} & c_{3}\end{array}\right\vert$$

13.3.6 Natural Unit of Volume

This volume formula establishes the unit of volume as the smallest non-empty tetrahedron by volume with vertices occupying integer coordinates. The following Tetrahedral vertices form an example of this unit tetrahedron in one of four possible orientations (since it is irregular):

$$\begin{array}{c} \left(\begin{array}{cccc} 1, & 0, & 0, & ... ...gin{array}{cccc} 0, & 0, & 0, & 0\end{array}\right)\end{array}$$

The smallest integral regular tetrahedron by volume with vertices occupying integer coordinates consists of four joined unit tetrahedra, one in each possible orientation. The following Tetrahedral vertices form the smallest regular tetrahedron, which has a volume of 4 (and edge lengths of 1):

$$\begin{array}{c} \left(\begin{array}{cccc} 1, & 0, & 0, & ... ...gin{array}{cccc} 0, & 0, & 0, & 1\end{array}\right)\end{array}$$

The smallest regular octahedron with vertices occupying integer coordinates consists of 16 joined unit tetrahedra. The following Tetrahedral vertices form the smallest regular octahedron, which has a volume of 16 (and edge lengths of 1):

$$\begin{array}{c} \left(\begin{array}{cccc} 1, & 1, & 0, & ... ...gin{array}{cccc} 0, & 1, & 1, & 0\end{array}\right)\end{array}$$

Bibliography

1

Fuller, R. Buckminster, in collaboration with E. J. Applewhite, Synergetics, New York, Macmillan, 1975, http://www.rwgrayprojects.com/synergetics/synergetics.html

2

Jarmusch, Darrel, Posted in March 13, 2000 in The Math Forum. Retrieved January 2002 from http://mathforum.org/epigone/geometry-college/dwoxgloncron, also http://euch3i.chem.emory.edu/proposal/www.teleport.com/~pdx4d/jarmusch.html

3

Urner, Kirby, An Introduction to Quadrays. best viewed at http://www.grunch.net/synergetics/quadrays.html.

4

Ace, Tom, Retrieved January 2002 from http://www.qnet.com/~crux/quadray.html

4

Hamilton, W. R., On Symbolic Geometry, The Cambridge and Dublin Mathematical Journal, 1846-1849. Retrieved in Postscript format January 2002 from http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/SymGeom/SymGeom.ps.

5

Grassmann, H., Die Ausdehnungslehre, Berlin, 1862.

6

Clifford, W.K., Applications of Grassmann's extensive algebra, Am. J. Math., 1:350, 1878.

7

See http://mathworld.wolfram.com/ContinuedFraction.html.

8

See http://homework.jhax.net/math/Pythagorean_Triples.jsp

9

See http://homework.jhax.net/math/Natural_Triples.jsp

10

See http://groups.yahoo.com/group/synergeo/message/626.