In Greek, geo means the place, and metrein means measuring.
A geometry is a set of axioms (accepted truth) and transformations that describes the legal movement possible in a place.
Any changes in the axioms changes the geometry. For example change one or two axioms of euclidian geometry gives you the geometry used by Einstein.
By changing a place in another you can compare them, by comparing them you measure.
There are no projections in one dimension In order to get dimensions you have to give your geometry new rules that’s called adding dimensions. Normally this operation is recursive. So once you add a dimension you can add as much dimension that you want.
Transformations and projections are operations that transform a place in antoher place. If it there is a reciprocal non deformative operation it is a transformation if you lose information in the process it is a projection.
Projection are destructive operations. Transformations are lossless operations in terms of operations. Projections can reduce the dimensions of the studied place (topos in greek).
Symmetries are transformations that given a set of rules leaves certain properties as-is.
Measuring is the magic that transforms geometrical rules in number.
It comes with three items that must consistently be defined :
The algebrical distance must be:
The reductor is the geometrical counterpart of the distance and define how you scale from a unit distance to a given topos. It answers to the question: given my multi-informational space how do I obtain a 1D problem? It may be a projection.
A projection is an operation yielding linear results, whereas a reductor yields a non linear result.
A constrictor is tightly coupled with the two items it defines the accepted transformation. From this will come the distance you use.
Example
A Manhathan geometry implies you move in a squarish universe (constrictor). Then the reductor is counting all the vertices you have to walk in this universe to go from place (topos) A to B using the «shortest path», and finally the distance is Sum(abs(vertices)). Defining the shortest path is equivalent to define two places known as :
A non euclidian geometry appears every time you play with axioms, accepted transformations or measures.
Since numbers, thus distance boils down to comparing to unit places, a measure is always a comparison with unit measure. There are no absolute numbers. That might be the reason I hate numerologists and astrologists. People taking for certain the outcome of numbers when numbers are only ratio to a unit place is like someone telling me: 8 is a great number after I have been beaten to the pulp once and telling me to get beaten 7 times more (I know chinese takes 8 for a lucky number).
A point is always a segment (the smallest place possible) given an arbitrary origin.
Example: the date of today is the number elpased since the arbitrary so called birth of Christ. Well, he was born sooner since he add is christmas gift on a 25th of december just very near to the proto christian religion begining of the year (21st of december) when the earth begins its trip back from the furtherst point to the sun resulting in the lenghtening of the day in the northen part of the earth.
Dear christians if earth is flat explain to me why Australian babes are in bikini when we great the new year in polar clothes in Montréal? To explain this come up with a good geometry that bits Terry Pratchett most hallucinated novel, it will be entertaining.
I believe time should be included in geometry as a side effect of projections. You have time once you cannot go back. Thus any operation that given a place has more than one predecessor defines a time (irreversilbility).
A time line is a serie of projections or reductor that don’t let the invariant of the last projections be the invariant of the next projection. The serie of projection can be continuous. It could be called an expander or a cruncher. Pick your name.
A geometry is virtual and reality is boring. However, what is described in the virtual world might come true given either an awful lot of good premisces and hard work to keep things consistent with observations or a short sight. Most people prefer short sight by lazyness.