II.   Divine Systems

The PRIMARY RULES of the ELEMENTAL CONCEPT of GEOLEX of the GEOLEX FOUNDATION CLASS of the EIKOS Language System are:

1. GEOLEX is a FOUNDATION CLASS of ELEMENTAL CONCEPTS associated with the study of principles and rules of definition by which GEOMETRIC SHAPES are defined, classified and used in valid statements defined by the EIKOS Language System.

2. GEOLEX is constructed of forty-six [46] ELEMENTAL CONCEPTS including: GEOLEX, POINT, DIMENSION, CONSTRUCT, PATH, LINE, SHAPE, SURFACE, DISTANCE, AREA, MEASURE,VOLUME, ANGLE, VECTORS,DIRECTION, 2D CONSTRUCTS, 3D CONSTRUCTS, TRI 2D, QUAD 2D, CIRCLE 2D, ELLIPSE 2D, PENTA 2D, HEX 2D, COMPLEX 2D, SPHERE 3D, PRISM 3D, PYRAMID 3D, CUBE 3D, CYLINDER 3D, CONE 3D, ELLIPSOID 3D, PARABOLOID 3D, TORUS 3D, COMPLEX 3D, PERFECT 2D CONSTRUCTS, IMPERFECT 2D CONSTRUCTS, PERFECT 3D CONSTRUCTS, IMPERFECT 3D CONSTRUCTS, FRACTALS, MORPHOLOGY, TIMEFRAME, MODEL, NATURAL GEOMETRIC SHAPE, GEOMETRY, SYMERICS and AXIOMATICS.

3. All elements defined by EIKOS, LOGOS, NUMERICS, UNISET and GEOLEX may be expressed as a FORMULA of one or more VARIABLES defined by the rules of SYMERICS.

4. All FORMULA defined by the rules of SYMERICS must adhere to the Ucadian Semantic Classification System standard of declaration of DIA (statements of DA) in their formulation and writing.

5. A DIA is a formal declaration of DA (OBJECTS as defined by the Ucadia Classification System and the Ucadia Symbols System).

6. The function of a VALID FORMULA (in terms of structure and flow of variables according to the rules of SYMERICS) may be expressed by only three (3) major types: LINEAR, MATRIX or MULTI-DIMENSION types.

7. All VALID FORMULA under the rules of SYMERICS may be further classified according to three (3) categories defining its primary FUNCTION and SCOPE of APPLICATION including NON-GEOMETRIC NON-NATURAL, NON-GEOMETRIC NATURAL and GEOMETRIC.

The PRIMARY RULES of the ELEMENTAL CONCEPT of POINT of the GEOLEX FOUNDATION CLASS of the EIKOS Language System are:

1. A NUMBER may be described as POINT in POSITION in two or three DIMENSION space.

2. Every POSITION is unique. No two POINTS shall ever occupy the same POSITION.

3. A POINT is uniquely aware of its POSITION in DIMENSION.

4. A POINT can only interact with immediate neighbour points.

The PRIMARY RULES of the ELEMENTAL CONCEPT of DIMENSION of the GEOLEX FOUNDATION CLASS of the EIKOS Language System are:

1. DIMENSION is the existence of one POINT having UNIQUE POSITION relative to more than one other POINT.

2. DIMENSION cannot exist without at least two (2) POINTS.

3. DIMENSION may also be defined as the POSITION observed by the POINT relative to other POINTS.

4. TWO DIMENSION is when less than six (6) POINTS exist in relation to one another.

5. THREE DIMENSION is when six (6) POINTS or more exist in relation to one another.

6. Within the EIKOS Language System there is no such valid concept as more than THREE DIMENSIONS.

7. SPACE is a PROPERTY of POINTS in DIMENSION and not an independent OBJECT.

The PRIMARY RULES of the ELEMENTAL CONCEPT of LINE of the GEOLEX FOUNDATION CLASS of the EIKOS Language System are:

1. A LINE is any THEORETICAL or REAL LENGTH with 0 HEIGHT and 0 WIDTH between two (2) POINTS.

2. A LINE may be only STRAIGHT or CURVED

3. A LINE be CONTINUOUS or SEGMENT. CONTINUOUS is when either the START and/or END POINT are known. SEGMENT is when both the START and END POINT are known.

4. A STRAIGHT LINE is the shortest distance between two (2) POINTS.

5. PARALLEL LINES are two or more LINES in the same DIMENSION that do not meet however far they are extended.

6. A CURVE is any CONTINUOUS or SEGMENT LINE whereby the LINE is comparison to a STRAIGHT LINE is found not to be in PARALLEL.

7. An ARC is a portion of a CURVE.

8. PERPENDICULAR LINES are two LINES in the same DIMENSION that meet at a right ANGLE.

9. A STRAIGHT LINE that intersects a CURVE may be additionally known as a SECANT. The part of the SECANT contained between the points of intersection is a CHORD.

10. No STRAIGHT LINES exist in REALITY. Only in THEORY.

11. All LINES in REALITY are CURVED to some degree.

12. The Simplest example of STRAIGHT (Theoretical) and CURVED (Real) LINE in combination is a PERFECT Circle with the DIAMETER representing the STRAIGHT LINE and the RADIUS representing the CURVED LINE.

13. Within the EIKOS Language System there is no such valid concept as more than THREE DIMENSIONS.

14. SPACE is a PROPERTY of POINTS in DIMENSION and not an independent OBJECT.

The PRIMARY RULES of the ELEMENTAL CONCEPT of SHAPE of the GEOLEX FOUNDATION CLASS of the EIKOS Language System are:

1. All relationships between two or more POINTS create geometric patterns of LINES in two and three dimensional space called SHAPES possessing common and unique properties.

2. A SHAPE exists only when it has clear INTERIOR LINES.

3. No STRAIGHT LINES exist in REALITY. Only in THEORY.

4. The Simplest example of STRAIGHT (Theoretical) and CURVED (Real) LINE in combination is a PERFECT Circle with the DIAMETER representing the STRAIGHT LINE and the RADIUS representing the CURVED LINE.

The PRIMARY RULES of the ELEMENTAL CONCEPT of SURFACE of the GEOLEX FOUNDATION CLASS of the EIKOS Language System are:

1. A SURFACE is an enclosed AREA formed by three or more POINTS in forming a two or three DIMENSION SHAPE.

2. A Two DIMENSION SHAPE may have only one SURFACE.

3. A Three DIMENSION SHAPE may have more than one SURFACE.

4. The simplest two DIMENSION SURFACE is a CIRCLE with one SURFACE.

5. The simplest three DIMENSION SURFACE is a SPHERE with one SURFACE.

6. The total SURFACE AREA of a three DIMENSION OBJECT may be considered by calculating each SURFACE AREA as a two DIMENSION object and then adding the totals for all SURFACES.THA

The PRIMARY RULES of the ELEMENTAL CONCEPT of TRIANGLE 2D of the GEOLEX FOUNDATION CLASS of the EIKOS Language System are:

1. A TRIANGLE is a 2 DIMENION SHAPE with three (3) corners or VERTICES and three (3) sides or edges formed by the intersection of three (3) LINE SEGMENTS.

2. In 2 DIMENSIONAL THEORETICAL SPACE, any three adjacent non-collinear POINTS will form a unique TRIANGLE in DIMENSION.

3. All TRIANGLES may be defined by both the relative LENGTH of its sides and by its internal ANGLES.

4. For LENGTH: an EQUALATERAL TRIANGLE is when all sides are of equal length. An ISOSCELES TRIANGLE is when only two sides are of equal length and a SCALENE TRIANGLE is when all three sides have different lengths.

5. For Internal ANGLES: a RIGHT ANGLE TRIANGLE has one ANGLE of 90° . An OBLIQUE TRIANGLE has no ANGLE of 90°; An OBTUSE TRIANGLE has one ANGLE greater than 90°; and an ACUTE TRIANGLE has ANGLES all of which are less than 90°.

6. The SUM of LENGTHS of any two sides of a TRIANGLE always exceeds the LENGTH of the third side. This is called the TRIANGLE LENGTH INEQUALITY RATIO.

7. The SUM of Internal ANGLES of a TRIANGLE always add up to 180°. The sum of the External ANGLES of a TRIANGLE always adds up to 360°. When the ANGLES of at least two sides of a TRIANGLE collapse to 0° the TRIANGLE becomes LINE SEGMENT.

8. In any RIGHT ANGLE TRIANGLE, the square of the LENGTH of the hypotenuse (opposite side to right-angle) equals the SUM of the squares of the LENGTHS of the two sides forming the right ANGLE.

The PRIMARY RULES of the ELEMENTAL CONCEPT of QUAD 2D of the GEOLEX FOUNDATION CLASS of the EIKOS Language System are:

1. A QUAD is a 2 DIMENION SHAPE with four (4) corners or VERTICES and four (4) sides or edges formed by the intersection of four (4) LINE SEGMENTS.

2. In 2 DIMENSIONAL THEORETICAL SPACE, any four adjacent non-collinear POINTS will form a unique QUAD in DIMENSION.

3. All QUADS may be defined by both the relative LENGTH and combination as PAIRS of its sides and by its internal ANGLES.

4. For LENGTH: an TRAPEZOID is where no two pair of side are equal. A PARALLELOGRAM is where both pair of opposite sides are PARALLEL. A RHOMBOID is when adjacent sides are unequal and a RHOMBUS is when all sides are equal.

5. For LENGTH when adjacent sides are equal in pairs: A KITE with both pair of sides pointing out from the centre; and DELTOID with one pair of sides pointing inward towards the centre.

6. For Internal ANGLES: a RECTANGLE is a right-angled PARALLELOGRAM. A SQUARE is a right-angled RHOMBUS.

The PRIMARY RULES of the ELEMENTAL CONCEPT of CIRCLE of the GEOLEX FOUNDATION CLASS of the EIKOS Language System are:

1. A CIRCLE is a CURVE in DIMENSION for which all of its POINTS satisfy in DIMENSION a DISTANCE equidistant from a given CENTRE POINT. This LINE is also known as the CIRCUMFRANCE.

2. A CIRCLE is the simplest example of STRAIGHT (Theoretical) and CURVED (Real) LINE in combination with the DIAMETER representing the STRAIGHT LINE and the CIRCUMFRANCE representing the CURVED LINE.

3. A STRAIGHT LINE that intersects a CURVE may be additionally known as a SECANT. The part of the SECANT contained between the points of intersection is a CHORD.

4. The longest CHORD of a CIRCLE which passes through the CENTRE is known as the DIAMETER.

5. A segment of the DIAMETER whereby one end is the CENTRE and the other touches the CIRCUMFRANCE is the RADIUS.

6. The relationship between DIAMETER and CIRCUMFRANCE is always expressed by the ratio 1 /pi.