# The Usual World

In order to compare dimension-related experiences between different
configurations, first let’s categorize dimension-related experiences and the
types of descriptions they can be given for our usual dimension-related
experiences — the **usual world**.

At the scale of our everyday lives, in the usual world, we experience a
configuration of exactly 3 spacial dimensions and 1 temporal dimension. We know
that the usual configuration has 3 spacial dimensions because there are only 3
independent **directions** (**vectors**) (I will use “direction” and “dimension”
roughly interchangably) in space for movement (forwards or backwards):

- There are no more than 3 spacial dimensions because: if you posit any 4th
direction for movement, then we can show that any destination that can be
reached by moving in that direction can be reached by moving some amounts in
all of the original 3 directions (refer to this as the original 3 directions
**simulating**the 4th direction). This implies that any spacial location can be described relative to your original location by 3 numbers (the amount to move in 3 dimensions). - There are no less than 3 spacial dimensions because: if you posit just 2 directions for movement, then we can show that there is a destination that cannot be reached by moving any amounts in those directions.

Of course, the 3 directions are not unique — they need just be **independent**
(i.e. no selection of the directions can be simulated the directions not in the
selection; this is referred to as **linear independence** in linear algebra).

Notice that some choices of directions seem different than others. Usually, a
coordinate system is presented with **orthogonal** (at 90 degree angles to each
other) directions for each dimension. And configurations where one direction is
switched around for another seem different too (this will cause the tuples of
numbers describing relative spacial positions to be different, for example). But
are these changes *really* different, or are they different ways of talking
about the same thing?

# Invariant Transformations

To answer this question, we need to specify what mean by “different” and “same”. What we want to achieve in the answer is that dimensional configurations that are “the same” will be indistinguishable from an inside perspective (like ours). For each of the following configuration transformations, suppose that there are two universes — ours and one just like ours except the transformation has been applied — and further suppose that you are placed at random in one of those universes. Would you be able to say with odds better than 50% which universe you are in?

- Flip over some direction
- No. TODO: explain

- Skew two directions to be at less than a positive angle less than 90 degrees
- No. TODO: explain

- Scale all directions by a factor of 2
- No. TODO: explain

- Remove a direction
- Yes. TODO: explain

- Add an orthogonal direction
- Yes. TODO: explain

How do these results follow from intition? From an outside perspective (like how we are describing them) these configurations are seemingly different. But from an inside perspective, importantly, there is no external frame of reference. The flipped universe looks just like the usual universe from the inside because the dimensions are perfectly symmetrical. As for the skewed universe, TODO.

This goes to show that there are certain **invariant transformations**, or
**invariants**, to avoid taking into account when categorizing dimensional
configurations; if a configuration is invariant under a transformation, such as
flipping or skewing, then all variants of that configuration must be of only one
category as they are indistinguishable from an inside perspective.

On the other hand, if a transformation is applied to a configuration under which it is not invariant, such as removing or adding a direction, then the original configuration should be in a different category from the transformed configuration; the original configuration and the transformed configuration are distinguishable from an inside perspectve.

# Variant Transformations

What other kinds of transformations can be done to configurations that produce
new categories? i.e. variant **transformations** or **variants**.

As we have already seen, adding or removing a dimension are variants on our
usual configuration. But what *kinds* of dimensions are they? And can we change
the 3 variantly change the 3 dimensions of the usual condifuration as well?

In the usual world, a spacial dimensions is thought of as extending forwards an backwards infinitely. Consider the same question of whether an inside perspective could distinguish the usual world from the transformed one when each of the following transformations is applied to some dimension(s):

- A dimensions is circular, such that moving along it some distance eventually
returns you to your original location
- Yes (possibly unfalsifiable). If you move along a dimension and end up where you started, then you have proven that the dimension is circular. If you move along a dimension any finite distance and do not end up where you started, you have not proved that the dimension is not circular.

- A dimension is finite, such that you may only move along a dimension (forwards
or backwards) a finite distance.
- Yes (possibly unfalisifiable). If you move along a dimension eventually cannot move in that direction any more, then you have proven that the dimension is finite. If you move along a dimension any finite distance and do not ever become unable to move in that direction any more, then you haev not proved that the dimension is not finite. -