Spencer Heath Archive

Item 62

Penciled by Heath on notepad paper. Includes several figures

No date

 

 

 

     From a point may be drawn in the same plane any number of lines forming any number of equal angles  —  i.e. equal diverging lines from a point

 

     When only two such lines are drawn the two equal angles are maximum (180 degrees)  ²/₂ =1=maximum

 

     When 3 such lines are drawn their angle (divergence) is ²/₃ the maximum.

 

     When 4 such lines are drawn their angle (divergence) is ²/₄ the maximum.

 

     When 5 such lines are drawn their angle (divergence) is ²/₅ the maximum.

 

     6    2/6

 

     7    2/7   etc.

 

                        /One-and-a-half pages of penciled                                diagrams here require photocopying./

 

 

A point is Reality at its least manifestation. Reality is triune — of Mass, Motion and Duration.

 

     Reality — Energy — is always composite of three insepar­ables — Mass, Motion, Duration — variously organized with respect to each other — among themselves.

 

     The least unit in which energy is manifested is in physics a quantum, in geometry, a point. Nothing smaller can be experienced.

 

     All quanta are in juxtaposition, all points contiguous.

 

     A series of quanta in juxtaposition is a finite quantity of energy.

 

A series of points in juxtaposition is a finite line.

 

A series of quanta in which mass is maximum is a structure (Solid).

 

In which motion is maximum, a ray.

 

In which duration is maximum, a life form.

__________________________

 

 

     A series of contiguous points in which the number between /?/ the first and last is maximum is a volume.

 

     A series of contiguous points is a line. Two or more lines that have any point in common are said to intersect.

 

     Two points and the least number of contiguous points connecting them constitute a straight line. Any other line is a curved line.

 

     A plane is an indefinite series of points having unbroken contiguity, in which all points are connected by straight lines and in which all lines that converge inter­sect.

 

     Two lines that are not equidistant throughout are said to converge. When the shortest straight lines connecting points in one with the nearest points in the other are of unequal length

 

    Equidistant when every point in one is separated by an equal number of points from a corresponding point in the other.

 

     A closed series of points and all points enclosed by it constitute a figure.

 

     A figure every point within which is connected by a straight line to all other points is a plane figure.

 

     Straight lines within the same plane figure that do not intersect are parallel lines.

 

     An angle is the divergence between two straight lines within or

 

     Two straight lines are equal when the end points in each are connected by the same number of points.

 

     Lines that converge are considered as extending towards a single point of intersection or towards two points in the shortest line between them. When considered as extend­ing away from such point or points they are said to diverge.

 

     Two lines that extend oppositely from the same point have maximum and equal divergence. Three lines that extend from the same point with equal divergence from each other have ²/₃ the maximum divergence. Four lines have ²/₄ maximum etc.   

 

     Any number of lines diverges from one point in the same plane. Any number of planes contains the same point

 

    Lines are not dimensions but directions. A line of soldiers is not a dimension. Dimensions are numbers.

 

     Lines are indefinite in extent and therefore have no lengths. Only parts of lines have dimensions, lengths — the part extending between any two points in it.

 

     The length of any part of a line is the number of points (or other units of measure) contained in that part. This number is usually spoken of as the length of the line, mean­ing the length of the part being considered.

 

     Directions are divergent from a point and indefinite in number. Dimensions are cumulative from a point and like­wise indefinite in number.

 

     Divergence expresses the unlikeness of complete, indefinite lines. Dimension expresses the likeness between parts of lines or of the same line

 

     A line of points is not a dimension any more than a line of soldiers is a dimension. The dimensions are the numbers (of points or of soldiers) between any two points in the line.

 

     It is nonsense to talk about three dimensions or four. There are any number of dimensions. It is equally nonsense to talk about three directions or divergences. There are any number of them.

 

     But there is a fixed maximum of direction divergence between two lines or between two planes. The maximum is called 180 degrees; then it returns on itself to zero. It is a reversible /?/ phenomena. There is no such maximum for number or length. We are therefore disposed to rate diverg­ence in terms of fractions of its maximum.

 

    Perhaps because things tend to fall in a direction midway between that of two horizons; perhaps because bricks or stones stay piled better with faces 90 degrees apart — whatever the cause — we early acquired the habit of think­ing as though there were only three directions (miscalled dimensions) and relating them exclusively to the cube; quite a worthy procedure, no doubt, but certainly not the only possible.

 

     Had our remote forebears found triangles and tetrahedrons more useful no doubt they would have done more sixty-degree thinking and even we would by now be more acute in our conceptions and less blocky in our heads,

 

     We would have invented the same arithmetic, but our play-blocks in practical architecture, monuments /?/ and geometry would have been tetrahedrons instead of hexahedrons and we might have become even more than ever confused into thinking our divergences were our dimensions and have even a tougher time than we now have in trying to find a fourth or fifth somewhere in a demoniac or metaphysical shadowy world.

 

     Let’s see what would have happened:

 

     A triangle with equal sides would have made a nice plat or field and fitted up to other triangles just as neatly as squares. And four would fit nicely together to make one of only double size on the side. Also by connecting opposite mid-sides any such triangle would divide up into four with only half as much side. The tent-sides patterned from the floor would serve as four walls and roof combined — a four to three gain, for the simple life.

 

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Commentary by Alvin Lowi, Jr.

III, IX D.1, 2, 3

 

SUBJECT:  Geometrical Analogs: The “Geometry” of Events

The point in geometry as the analog of the quantum in physics. Quanta are juxtaposed, points contiguous. Juxtaposed quanta are energy — contiguous points, lines. Angles and divergences.

Geometrical structures (concepts) in terms of contiguous points: lines, angles, solids, volumes, surfaces, figures, etc.

Lines as directions — not dimensions. Dimension relates, direction differentiates.

Dimensions and Directions — a multiplicity with limits.

Rectangular thinking

 

REMARKS:  Good stuff on space, form and dimensionality — very imaginative.