Set Euclidean Coordinate System

Euclidean space, and spherical space.

It's a definition really of Euclidean space.

Euclidean space, populated uniformly with some kind of sources.

It's a bit like this imagine that we'd only ever encountered Euclidean space.

You all have a sense of what a flat space is, Euclidean space is.

low distances from us makes us still nearly Euclidean. So the lines will be deviating.

In a Euclidean space, the further away something is, the smaller it's going to look, always.

The discovery of hyperbolic space ushered in the field of mathematics that is called non Euclidean geometry.

It boils down much to the number of which cell steps but for the Euclidean distance to a target location.

In a simple Euclidean non expanding space, that angle will be the size of the ruler divided by the distance.

Thus, the number will scale as a flux to 3 2 power, and that is what we call Euclidean source counts.

But then the further out we go the relativistic effects become more important and the line peels off from the Euclidean asymptote.

Things are close to Euclidean and therefore the counts will be a asymptotically going to the straight line with the slope we just derived.

So in simple, Euclidean, non relativistic universe, the flux would be luminosity divide, divided by the area of the sphere with a radius from here to there.

As you go through this video you can see how A star planning with a simple Euclidean distance heuristic is able to find a path to the goal.

The point is, however, that both numbers of sources And their fluxes depend on cosmological, but as it turns out for all reasonable cosmological models the slopes can only deviate in one way from the Euclidean.

If the sources were brighter in the past, than those which were really faint by coming closer the words the Euclidean slope line, and so that will tend to push it above the no evolution cosmological

Here as well as in the other plots, all of the curves are really close together at very low redshifts, when things that are asymptotically Euclidean. So that's the distances, what about the look back time?

There are two reasons why sources go fainter, the first one is the luminosity distances the one plus z factor if you recall because of that alarm, regardless more or less regardless of cosmology the, The sources will be fainter in an expanding universe then they would be in a equivalent euclidean space.

I'll see that this is equal to that on its own. but what's so linear about them? what makes them look like a line? to realize why they're linear, you have to make this jump René Descartes made. because if you were to plot this, using cartesian coordinates. on a Euclidean plane.

This is spatially flat model, good old Euclidean geometry, it follows the spacial relativity and it is realized in Sanskrit there no matter at all, it's any empty universe, just specs, so you can think of it as galaxies having no mass of their own just being test particles to show us how the coordinates are expanding as you will see how, Hubble's law follows immediate and directly from it.

But before Descartes is generally viewed. that geometry was euclidean geometry. and that's essentially the geometry that you studied in geometry class in 8th or 9th or 10th grade. in a traditional high school curriculum. and that's the geometry of studying the relationships between triangles, and their angles. and the relationships between circles. and you have radii and then you have triangles inscribed in circles and all the rest and we'll go into some depth in that in the geometry playlist.