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

742 Types 9, 11, 12, and 13: The Rice pentagons

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

Start my 1-month free trial

742 Types 9, 11, 12, and 13: The Rice pentagons

“

- [Instructor] In this movie we'll look at types nine, 11, 12, and 13 of the guaranteed to tessellate perfectly irregular convex pentagons. And to see what they look like, I'll go ahead and click at this down pointing arrow head in the bottom left corner of the screen here inside the illustrator, the one labeled Artboard Navigation and I'll scroll down to this guy right here, Type 9. So again, types nine and 10 are out of order for reasons I do not know. But I'll just go ahead and choose Type 9 which happens to be on Artbard 16 in order to see the first of four pentagons discovered by Marjorie Rice in 1977. And this is a woman by the way, who had nothing more than a high school education and yet she was able to figure out more of these than anyone but that first guy Karl Reinhardt back in 1918. Alright so once again the angles up here in bold and the sides appear in italics, and we've got these rules. So, where the angles are concerned, 2A plus C is the same as D plus 2E which adds up to 360 degrees. Which as we'll see proves to be a little bit challenging. And once again, all of the sides except A are the very same length. So B equals C which equals D and E which means that these sides can abut each other. And to see what that looks like, I'll go ahead and switch back to my layers panel and I'll turn on my patterns layer so that we can see that the tile ends up repeating a total of eight times, which is why we have eight different colors on screen. And to see how I assembled things, which took a little bit of effort, I'll turn on a couple of more layers here. And notice this is the original shape colored in light blue, I went ahead and flipped it across the vertical axis so that amounts to a horizontal flip and then I rotated it 46.63 degrees which happens to suite this specific shape quite nicely. And then, I went ahead and took both of these shapes and rotated copies of them 78.85 degrees. And so why these values are working, I really don't know, I just know that they ended up making everything fit perfectly. Then, I took all four of these shapes and I flipped them across the horizontal axis so in other words, a vertical flip and then I rotated them all 125.48 degrees in order to create these orange shapes right there and then I dragged around a bunch of copies in order to create this manual pattern. Alright, I'm gonna go ahead and turn off that patterns layer and I'll press Shift Page Down in order to advance to the next Artboard which takes us from Type 9 to Type 11, once again created by Marjorie Rice. This time around we have more of an elongated shape, in which angle A is exactly 90 degrees. So we've got a right angle at this location. Then 2B plus C equals 360 degrees and C plus E equals 180 degrees. Which by the way, may possibly exceed my ability to visualize at this point, but we have side D equaling side E, so they're exactly the same length, which also happens to equal 2A plus C, which is why I've colored side A in orange and you can see there's A against E. There's another A against E, so a second instance and then we have C colored in green in between. And we're seeing the same thing against side D. If you're interested in how this shakes out, I'll go ahead and turn on the patters layer once again and we end up with these kind of fish faces biting each other. Once again, I'm duplicating the base tile a total of eight times. And to make that happen, I went ahead and took that first shape and I flipped it across the H axis. So quite obviously we have a vertical flip and then I took those two shapes and I flipped them across the vertical axis in order to produce a horizontal flip. So at least where these four shapes are concerned, it's pretty easy. And then I took block one, which I'm calling these two shapes right here, and I rotated them negative 58.79 degrees in order to fit these two shapes into this location. And then I took these two block two shapes and rotated copies of them by the very same amount, negative 58.97 degrees and I ended up with this more or less perfect fit right here. Alright, I'll go ahead and turn off those layers and I'll advance to the next Artboard so we can see Type 12, which has some very similar rules. So all the angles shake out the same, that is to say just like before. So notice this is Type 11, this is Type 12. A is equal to exactly 90 degrees, so a right angle, 2B plus C, whatever they are equal 360 degrees and then C and E, which can vary as long as they together equal 180 degrees. This time D, this side right here, is equal to 2A, so two times A, which is colorized in red. The reason we're seeing green and orange against side D is because it's also equal to C plus E, so C is green and E is orange. Which means all these sides compile up against each other as witnessed in this pattern right here. And so we have these kind of fish faces once again but they're biting each other at different angles. So this is Type 11, just to refresh your memory and this is Type 12. And so to assemble this pattern I came up with this approach, you might try things differently. But I started with this original shape, I flipped it vertically, so across the horizontal axis and then I flipped them both across the vertical axis, so horizontally, which is to say I started things in exactly the same way but then I took the entire group, so I didn't approach it one block at a time like before, I took all of the shapes and I flipped them across a horizontal access, so vertically and then I rotated them 62.53 degrees, ended up working out. Alright, I'll go ahead and turn those layers off and we'll advance to the final Marjorie Rice pentagon which is Type 13 right here. This time the angles shape out differently. So we've got B equals E equals 90 degrees so both B and E are right angles. And then, 2A plus D equals 360 degrees, whatever on that one, but where the sides are concerned D equals 2A which also equals 2E. So in other words, sides A and E are exactly equal to each other and then if you multiply either one of them times two you end up with side D. And that looks pretty regular as we're seeing right here, so notice that we're getting some extremely horizontal and vertical alignment out of our repeated shapes, which one again, add up to eight. So in other words, I took the first one and duplicated it a total of seven times. Which looks like this right here. So here's our original shape, I flipped it vertically across the horizontal axis, I flipped both of them horizontally, so across the vertical axis and then I took all four and all I had to do this time was rotate them exactly 90 degrees, presumably because we've got those right angles. Now you may not really care for this perpendicular orientation, in which case, just go ahead and click anywhere inside that pattern in order to select the entire thing and then go up to the Effect menu, choose Distort & Transform and choose the Transform command which will allow us to rotate the pattern dynamically. And so I'll go ahead and choose a command, if you get this alert message telling you that I've already applied some transform effects, just go ahead and click Apply New Effect in order to bring up the Transform dialog box. Then, turn on the preview checkbox, very important, so you can see what you're doing, click in a rotate angle value and press shift up arrow, maybe a couple of times in order to rotate the entire pattern 20 degrees. Now these are actually path outlines, so go ahead and leave the transform objects checkbox turned on at which point you can click OK in order to apply that effect. And there you have it, four more types added to the pantheon of perfectly tessellating, irregular convex pentagons as discovered by Marjorie Rice, a mom who had no more than a high school education, doing much better than I would with my Bachelor's Degree in response to a Scientific American article back in 1977.

Contents