On rotations as continuously-reflected translations

Image of PDF file - single page

rotatn_is_reflected_translatns.pdf is a pdf of a little diagram I created this morning, where I muse with the ideas of transformations and how we can express Rotations (and circularity or curvature) using “not” – the ‘omni-dimensional’ Reflection (how would you describe a zero-yet-infinity-dimensional thing? “multi-“? or “trans-“?).

I don’t know why I called those vectors in the diagram ‘rays’  – they’re vectors which describe a translation transformation.

If this is right, why would we need ‘curved’ non-Euclidean spaces?



This zero-D Reflection is a strange bird…

The very existence of a boundary to separate a space in two gives rise to a higher dimension in which both can exist. In zero dimensions, a Thing bounded by a Reflection (“Not”), then (and I know the words are dangerously wrong here) has a kind of ‘inside’ which is not ‘outside’. Strictly speaking, this ‘inside’ is no longer even in the apprehensive realm of whatever is ‘outside’ – precisely because of the boundary which ‘shuts the door’ to the ‘inside’. It is clearer in 2D and then you can roll-back the analogy to 0D later:

Think of a circle – an enclosed section of 2D space. To a 2D observer, that circle is closed and has a boundary. Because it is closed, then the 2D observer might conclude that the circle has an ‘inside’ – but has no way of getting to it, unless one of two actions are taken:

1) Using a special 2D scalpel, the 2D observer ‘cuts’ the circle open. By doing this, it has reduced the circle to  a 1D line – an object in a lower dimension than the observer, and so both ‘inside’ and ‘outside’ can be observed (though arguably both ‘inside’ and ‘outside’ are lost, the moment the circle was ‘cut’)

2) Trying to ‘step over’ the boundary. Like trying to jump a fence – the only way to do this is to briefly traverse the 3D realm, only to land ‘inside’ the circle. But because the observer is 2D, now ‘inside’ the circle, then the observer loses all apprehensive ability of ‘outside’ the circle. It’s either-or.

The Reflection in a dimension, I’ve said elsewhere, has a special property in that it is a Half-Rotation in a higher dimension. So to pass from ‘inside’ to ‘outside’ (and vice-versa) you have to be subjected to a higher-dimensional Half-Rotation. But you’ll start and finish in the original dimension. What if you wanted to stay above 2D? Well, you’d have to ‘not’ complete the Half-Rotation. You’d have to do a Quarter-Rotation of sorts. This is obvious when we consider that the third axis of 3D is 90° perpendicular to the other two, but it’s also beautiful when we consider that some Thing both Is and Is Not in a higher dimension. So back to zero-D, where what can be said of a Thing is that it “Is” or “Is not” – in 1D then you can ‘see’ both.

Consider this line: —–IS—–<NOT>____ISN’T____<NOT>—–IS—–

The ‘region’ bounded by both those ‘not’ Reflections, is invisible (incomprehensible) to a 1D observer. We see it because we’re in a higher dimension.