David… Are you a scientists?
I can think of many more ways on the slate in my head, but I’m sure a computer program will find many more.
Let us assume that the five successive events occur 72 degrees apart around the, we’ll say circular, orbit (n.b. 5 * 72 = 360). With out ‘lifting pen from paper’, this would trace out a pentagon as shown in the case N=1 in the diagram.
Well, what if they occur 144 degrees apart; ah, here’s our beloved pentagon, drawn in the connected fashion you mention (see N=2).
Hmm, what about (3*72=) 216 degrees apart; well that makes the same continuous pentagram shape only the other way round (N=3).
(4*72=) 288 degrees? Okay, we’re back to the pentagon again (N=4).
Now, 5*72 is 360 degrees, so we don’t get anything, every event would occur in the same point of the orbit (N=5).
Can we go further? Well 6*72 = 432 degrees, which is one complete circle + 72 degrees, so we just get the same as the N=1 case, restarting the pattern (N=6).
Ergo, if it draws a pentagram, it must do so without ‘lifting pen from paper’. So, no, I am not amazed by fairly elementary maths.
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