The aspect


page 6 of 8


The relationship between chord and radius


The square root connection

As we have discovered before, reality consists of multiple dimensions and the analogical mode of thinking can lead us from the physical world into those other dimensions (3.2). In those other dimensions, we can catch sight of interconnections behind physical appearances which give those appearances a deeper meaning. Each of these realms stands for a field of experience. In our discussion of the great circles in three-dimensional space, we saw that these fields invariably are perpendicular to one another, and we also determined through which gateways one area becomes accessible from another (9.3).
In our present discussion of the aspects, once again we encounter two phenomena, the chord and the radius, that are perpendicular to one another, implying that each of them will exert its influence in a different dimension. Yet they turn out to be related to each other through a square root connection.
In order to better understand which gateway is meant here, I will dwell a little more on this algebraic relationship.


Raising to a higher power

The counterpart of extracting a number’s square root is raising it to the square or to a higher power. Raising the length of a line segment, expressed as a numerical value, to a square, we make it longer. This quantitative squaring of a line segment elongates it further into the field where it is situated. For example, a line segment of 3 cm will become 9 cm in length in its squared form. This operation expands the line segment in the sense of ‘more of the same’. There is no quality added to it nor taken away from it.


Quantitative and qualitative

However, when we square the length of a line segment in a direction perpendicular to it, we get a surface. For example, from the line segment of 3 cm we will obtain a plane with a surface of 9 cm². In this operation, a second dimension is added to the first. The term ‘raising to a higher power’ refers to this areal expansion. In this case, the increase refers not to an elongation but to an elevation from a previous dimension.  
A subsequent elevation or raise to a higher power will open up the area perpendicular to both preceding areas; this second elevation will result in a cube of 3 x 3 x 3 cm = 27 cm³. So by raising it to a higher power, a line will expand into a plane which in turn will expand into a space.

More dimensions

If we continue this operation, we will develop forms in dimensions that are beyond our three-dimensional world. Although we are not able to imagine those forms, this is no impediment in an arithmetical sense. This sequence can be simply continued, into infinity even. Although we do not know the laws governing those unknown realms (25.3.a), with each raise to a higher power, we expand our possibility to gain experience in the surrounding world and increase our freedom of movement.


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