From the course: Deke's Techniques (2018-2021)

743 Type 14: The Stein pentagon

From the course: Deke's Techniques (2018-2021)

743 Type 14: The Stein pentagon

- [Instructor] In this movie we'll take a look at type 14 of the perfectly tessellating irregular convex pentagons. And to see what that looks like, I'm gonna bring up the art boards panel by going up to the window menu and choosing the art boards command. And that way I can easily switch between a couple of art boards down here near the bottom of the list. So I'm gonna double click on type 14 in order to advance to this guy right here, which was discovered by a German math student named Rolf Stein back in 1985. And as I hope you'll remember by now, all the angles appear in bold right here, all five of them, and the sides appear in italics. And so notice we have some pretty familiar rules. A equals 90 degrees so we've got a right angle at this location. Two B plus C equals 360 degrees whereas C plus E equals 180 degrees. And if that does by any chance look familiar, it's because those are the same angle rules that we saw with type 11, which was created by Marjorie Rice. So the very same angle rules prevail. Though, as we'll see, they're more specific this time around. Notice that the sides where type 11 are concerned, d equals e equals two-a plus c. Compare that to type 14, where d and e still are equal to each other. But so are a and c this time around. So the length of side a is the same as the length of side c, and if you multiply either of them times two, you get d or e. We also have some very specific angle instructions that go beyond what I've listed here. And to see what that looks like, even though it's a little bit scary, I'm gonna return to the layers panel and I'm gonna turn on this layer right here, this demos layer. And notice that just for B, this is just one of the rules associated with angle B, the sign of B is equal to the square root of 57 minus three, all over eight, which happens to equal this value right here. So 0.568 and so forth. And then if you do an arc sign to figure out what B is equal to, it's 145.3383362 degrees and so forth. So rather than sharing all this information with you, which probably isn't gonna do you that much good, I've gone ahead and approximated each one of these angles, which is why they appear in beige. So I'll go ahead and turn off that layer. B is approximately 145.34 degrees, C is 69.32, D is 124.66, and you can probably read just as well as I can, that E is approximately equal to 110.68 degrees. And to see what that looks like, I'll go ahead and turn on the patterns layer here. And this time we still have these kind of fish faces, but we have these tails as well. And so to put those together, and again, you can approach your patterns in any way you like and still get the exact same effects. I went ahead and started with this first shape and I flipped it across the horizontal axis. So we've got a vertical flip. And then I took that second shape and I duplicated it, and then I rotated it 55.28 degrees in order to create this kind of fish tail or body or what-have-you. And then I took all three of these shapes and rotated them 180 degrees. So we end up with this kind of Pisces orientation. And so I'm gonna go ahead and turn those two layers off for a moment. I want you to see the ultimate difference between type 14 and type 11. So I'll go ahead and switch back to the art boards panel, which may be located at a different location on your screen, and I'll double click on type 11. So here you can see that I've duplicated that base tile a total of eight times in order to achieve this final pattern, whereas with type 14, we're repeating that tile just six times. And that is type 14 of the perfectly tessellating irregular convex pentagons first discovered by German math student Rolf Stein back in 1985.

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