Mathematically Speaking

It is possible to represent an angle (fraction of a turn) in Euclidean 3-space as a 2-dimensional object rather than as a simple ratio between two lengths.  This 2-dimensional object can be viewed mathematically as a value or quantity.  Because we can represent direction as a quantity this way, it must be possible to construct a geometry that uses this quantity as a metric, as opposed to using length as a metric.

We don’t know exactly how this is done yet because the algebra hasn’t been worked out sufficiently.  We do know that the area under the curve drawn by this 2-dimensional representation is a mathematical quantity.

This section contains preliminary drafts of ideas that are still under development.  More material will be furnished on these subjects as it is accumulated.

Before Normalization

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Before Normalization

Figure 4.2  Different Values of \upsilon

 

This is another graph by Strange and this one shows four different values:  \upsilon=\frac{\pi}{2.2}\:, \:\frac{\pi}{3}\:,\:\frac{\pi}{4}\:, \:\frac{\pi}{6}
The curves have different extents along the x axis, which is the \lambda angle.

In order to represent \upsilon as a quantity that expresses direction, the curves must be “normalized” by making the extent of the curves along the x axis equal to 1,  similar (mathematically and conceptually) to the way vectors are normalized in order to create a unit vector or direction vector.

Once the curves are normalized, they express direction as a function of direction, without any length.

After Normalization

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After Normalization

Figure 4.3  Normalized Values for \lambda

The curves representing \alpha=f(\lambda) have had the \lambda axis “normalized” and are rotated 90º to correspond to the orientation of the partial sine curve.

How We Got Here

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200 years ago, Napier and others wrote about the Law of Sines for spherical angles, expanding on work begun by the Arabs.  Approximately 150 years ago, a treatise on this branch of mathematics (Todhunter’s textbook “Spherical Trigonometry”) was published.  From then until now, this was considered to be the complete story regarding spherical trigonometry.  Nothing new has been revealed in this area of mathematics since those times.

When relativity became a popular study about 100 years ago, the existing (believed to be complete) understanding of spherical trigonometry wasn’t powerful enough to solve many of the new mathematical problems that were popping up.  So, they figured out a work-around where they could analyze the spherical surface while ignoring dihedral angles as being inconsequential, since another way had been found to solve these problems which yielded identical results.

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Bizarro Vector

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There exists a deeper understanding of what constitutes 3-space. Each orthogonal two-dimensional plane is most commonly defined as two lengths (or axes) perpendicular to one another, sure, but what does that really mean? The concept of orthogonality brings the concept of relative direction (or orientation) into the picture. This structure of relative orientation, when combined with length, is the defining characteristic of a two-dimensional plane.

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