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Here we have our two intervals on our single, purple line ‘permanently’ end-to-end linked by conscious construction. They are “daisy-chained”.  And we at once find that we have a new perspectivity, centered on M, making the interval PS equivalent to the black interval and, thereby, identical to the red and blue intervals, too!

We see that we can join N to S, and get a new equivalent black interval, and from P form a new perspectivity on it, to get a new, green interval. We see we can repeat this process, indefinitely, and develop a set of intervals in one line, all  identical with a single interval (PS), and so get our projective ruler.

There are exactly two places on the purple line, called “invariants”, beyond which this, “back-and-forth, up-and-down”, process of interval replication cannot go. In fact, they “end” the entire ruler.  Can you find them?

You can explore all this ruler's manifestations