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

741 Type 10: The James pentagon

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

741 Type 10: The James pentagon

- Hey, gang, this is Deke McClelland. Welcome to Deke's Techniques. Now this week we're still talking about those terribly exciting, irregular convex pentagons that tesselate. And so, you may recall from last week that back in 1968, a guy named Richard Kershner discovers types six, seven, and eight, and then goes ahead and proves that his three, along with the original five from Karl Reinhardt, are the only ones that are possible. Then along comes a guy named Michael Gardner, who does a column called Mathematical Games for Scientific American. He writes this whole thing up in 1975, and he fires the imagination of three people who prove Kershner wrong, and these people include Richard James III, who discovers the ninth type, which is called Type 10 for some reason. Then most extraordinarily, Marjorie Rice, who discovers four of these things, more than Kershner, and she has nothing more than a high school education. She discovers types nine, 11, 12, and 13, and then finally we have this one from German math student, what's his name, Rolf Stein, which was discovered in 1985. We're gonna start things off with type 10 coming at ya right now. All right, so if you're working along with me, go ahead and drop down to the bottom left corner of the screen here inside Illustrator. Click on this down pointing arrowhead that's labeled Artboard Navigation, and then scroll down the list until you come to Artboard 15, which is called Type 10. So why Type 10 is out of order when it was actually discovered a couple of years before Type Nine, I have no idea. But that's the one you want, and so here it is, Type 10, created by Richard E. James III in 1975. And it looks a little something like this. And so as you may recall, the five internal angles appear in bold type as letters A through E, whereas the length of each one of the sides appears italic, and so the rules here are that angle A is exactly 90 degrees, so it's a right angle. B and E add up to 180 degrees, and then B plus 2C equals 360 degrees. Now, while it's a little hard for me at least, to imagine how the angles fit together, the sides end up making a lot of sense. So notice that A and B are exactly equal to each other, and they're also equal to C and E added together, and so you can see that E, which is a longer one, is colored in green, whereas C, the shorter one, is colored in orange. And they happen to add up to sides B and A. And that's gonna end up looking like this, by the way, and so notice this greenish shape is kind of leaning backward, and so this is its side A right there, which is set against the original side E, and this rotated side C. And so that's how it ends up working out, with a central group of four, and then these other two shapes fill in the gaps in between. But the important thing to note is every single one of these shapes is the exact same pentagon, and so to see how I put that together, I'll go ahead and turn on these layers right here. I started with this original, so the light blue shape is always that shape I show you at the beginning, and then I went ahead and rotated a copy of it around this bottom left anchor point, exactly 90 degrees, a total of three times. I went ahead and created another copy of this light blue original, and I flipped it across the vertical axis, so in other words a horizontal flip, and then I rotated it negative 18.93 degrees, and I did so just because it happened to work out and it made sure that the shapes exactly aligned. Then I went ahead and rotated a copy of that shape 180 degrees in order to create this 6th and final iteration. And incidentally, if you're thinking that your pattern looks a little bit too perpendicular, then you can just go ahead and click on it with the black arrow tool, and you can rotate it. That is, the whole pattern, and the best way to pull that off is to apply a dynamic transformation, which you do by going up to the effect menu, choosing Distort and Transform, and then choosing the Transform command. Now I've already applied this effect a couple of times by now, which is why you may see this alert message, in which case, just go ahead and click Apply New Effect, in order to visit this dialog box right here, and now I'll turn on the preview checkbox, and I'll click on that rotate angle value right there in the center of the dialog box, and I'll press shift, up arrow, in order to increase the angle value to 10 degrees, or I could increase it to 20 degrees or 30 degrees, or what have you. Now, we are actually rotating objects as opposed to tile patterns, so just go ahead and leave this first checkbox selected, after which point you can click okay in order to apply that rotation. And if you ever change your mind in the future, all you need to do is go up to the Window menu and choose the Appearance command in order to bring up the Appearance panel, and notice that I've applied that Transform command one or two times, all to different effect. Now the top transform is the first one I applied, and the bottom one is the last one that we just applied, so if you want to change the angle of rotation, just click on that bottom guy like so and then I could take the angle value down to just, let's say 12 degrees, turn on the preview checkbox, and I end up with this dynamically rotated effect right here. At which point I'll just go ahead and click okay to accept that change. And that's how you work with the so-called Type 10 monohedrally tesselating pentagon first discovered in 1975. All right, now if you're a member of lynda.com/linkedinlearning, I have two follow-up movies, in which we take a look at pentagons types nine, 11, 12 and 13 from Marjorie Rice, as well as type 14 from Rolf Stein. If you're looking forward to next week, I'm gonna show you how to draw one of these pentagons absolutely from scratch inside Illustrator, and it's gonna be the most complex pentagon of them all, type 15. Deke's Techniques each and every week. Keep watching.

Contents