### Gervais Mulongoy

Shared publicly -A convenient mental tool we learn about later-on in school is the concept of a 'vector' (I'm honestly baffled as to why we don't learn about these earlier in school). A vector is just an arrow. But that arrow describes three important bits of information:

1) A line of action - the line along which it is pointing.

2) A magnitude - it's length, which can represent its 'intensity'.

3) A direction - which 'end' of the arrow the 'arrow head' is on.

Translations can be described using vectors. The arrow describes:

1) The line along which we're moving.

2) The 'intensity' with which we're moving.

3) The direction we're moving along the line.

Rotations can also be described using vectors. The arrow describes:

1) The 'axis' of Rotation.

2) The 'intensity' with which it is Rotating.

3) The 'direction' of the Rotation (clock-wise or anti-clockwise).

But we encounter a problem with Reflections. They cannot be described by vectors:

1) The 'line' of a Reflection? What does that mean in 'no space'? What does that mean in 3D space? Reflections aren't always 'along a line'.

2) The 'intensity' of a Reflection? How can you have a partial Reflection (remember we're talking about mathematical transformations, not the half-silvered mirrors etc.) - it either is or isn't Reflected.

3) The 'direction' - wait a minute...

Yes, 'direction' is actually a yes-or-no binary 'bit' of information which is saying simply that the vector is or isn't Reflected. So the 'arrow head' of a vector is actually conveying information about whether-or-not the transformation we're representing by the vector is Reflected.

So in 'no space', we can have one, and only one transformation - the Reflection. Interestingly, all that a Reflection really 'says' is 'yes' or 'no' - i.e. a Thing "is" or "is not". Furthermore, to cancel that transformation, you simply have to re-apply it, because a Reflection is its own opposite: if we say "No" and then we mean to change our mind, we could say "Not no; Yes". "not no" is the re-application of "no" to "no", thereby cancelling it.

For further 'validation' of how a Reflection can be possible in 'zero space' - or zero dimensions - we can look at the relationship between dimensions and transformations. I will propose another thought experiment: On a piece of paper, turned 'landscape' so you have room for this, draw a vertical line down the middle of the page. On one side of that line, draw a triangle (or any shape - triangles are conveniently easy to draw 'free hand'). The vertical line is your 'line of Reflection' - relative to which you want to reflect your hand-drawn triangle. There are two ways to do this:

1) You draw precisely-perpendicular lines from your 'Reflection line' to each vertex of the triangle. Measure the length of these lines, then draw exactly the same length lines on the other side of your 'Reflection line'. Then, simply connect the vertices of your newly reflected triangle. Laborious, imprecise, but it does work.

2) You fold your piece of paper along your 'Reflection line', keeping the blank side firmly on the table, and folding the drawn triangle over the top of it. Then trace your triangle on the other side (or poke holes through the triangle's vertices, or rub the page and try to get some graphite to transfer). With a clean fold, you're sure to get the exact reflection every time: quick and reliable.

Look at the second method: What have we done? We rotated the 2D object (the triangle) through 3D space. More precisely, we effected a half-rotation. Can the same be done with a 1D line segment? Yes: using the same page (let's be ecological here), draw a dashed-line anywhere on your page. On one side of it, darken a line segment (fill it in as a solid dark line segment). A little away from your segment, squiggle a black dot. This will be your 'Reflection point' (no longer a 'line' - reiterating why Reflections can't be represented by vectors). Fold the page through the point so that the dashed-line overlaps. Then trace your line segment on the other side. Voilà , you've just half-rotated a 1D object through 2D space and resulted in a 1D Reflection of your 1D line segment.

There's a pattern here: a half-rotation in n dimensions results in a Reflection in n-1 dimensions. So we can follow this pattern down to zero-dimensions: a 0D object half-Rotated through 1D space is a Reflection of that 0D object in zero-dimensions.

"Wait-a-minute" I hear you say. First, what the heck is any kind of rotation in 1D? Second, how the heck does that 'equate' to any kind of yes-or-no Reflection? Don't fret - here's another experiment to get your hands dirty:

Take a cardboard toilet paper tube and paint half side black (lengthwise, not 'around' the cylinder) and the other side white. Then roll the tube over a white background - better yet, have someone else rotate that tube in place against a white background while you watch, standing far far away. What you will see is the, admittedly 'rectangular', black dot appear and disappear, blinking in and out of existence... This is the Point being Reflected and then re-Reflected - in other words, our Thing for which we can say it "is" or "is not". For you, the black dot "is" and then "is not". Once again, we've seen that Reflections are viable transformations for a non-spatial dimension.

Reflections, furthermore, are not just 'zero-dimensional' - they're 'pan-dimensional'. Vectors, you may know (or will soon learn in school), can be represented in infinitely-many dimensions - except zero, as we've seen. But what we've also seen is that the arrow-head of a vector is the yes-or-no 'flag' indicating the 'Reflection status' of that vector. So _every_ vector, up to infinite dimensions, is implicitly constructed with a zero-dimensional Reflection. The reflection transformation is present in ALL (and NONE) dimensions.