These observations allow one to formalize the definition of reflection: a reflection is an involutive isometry of an Euclidean space whose set of fixed points is an affine subspace of codimension 1.
Through surveys researchers first determined that subjects perceive future events as being closer than past events, even if the events are equidistant.
According to Descartes, this is just as certain as it is inherent in the idea of a circle that all points of the circle are equidistant from the center.
The changes in the relative size of the orbits, and the lengths of the horns of ceratopsians as they age, are examples of what we call non-isometric autogenetic changes.
He inserts a screw into one end, and using a piece of dental floss, crushes the paper between the threads of the screw creating a series of tiny equidistant corrugations.
What's more, this function is actually linear, since it passes our visual test that any line of evenly spaced dots remains evenly spaced once it lands on the number line.
If we play some transformation and follow where all three of these vectors go, the property that greed lines remain parallel and evenly spaced has a really important consequence.
So in this case, the visual understanding for what linearity means is that if you have a line of evenly spaced dots, it would remain evenly spaced once they're mapped onto the number line.
I then took a piece of fine wood, and cut it like the back of a comb, making several holes in it at equal distances with as small a needle as I could get from Glumdalclitch.
The idea of grid lines remaining parallel and evenly spaced, that I've talked about in past videos is really just an illustration of what these two properties mean in the specific case of points in two D space.