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

739 Types 1–5: The Reinhardt pentagons

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

739 Types 1–5: The Reinhardt pentagons

- Hey gang, this is Deke McClellend. Welcome to Deke's Techniques. Now last week I introduced you to the 15 types of irregular convex pentagons that tesselate. So many syllables. And so I figured, a lot of you would like to know exactly what those pentagons are. Because after all, your workmates and your buddies, they've got their rectangular tile patterns, and their hexagons, you have those things too. But do they have pentagons? They do not. And so we're gonna start things off with the original five Karl Reinhardt pentagons which include this guy, type four, as well as this variation right here, and so much more. Here, let me show you exactly how they work. Alright now, if you have access to this exercise file, and you're working along with me once again inside Illustrator, then drop down to bottom left corner of the screen and click on this down-pointing arrowhead that reads art board navigation, and choose the fourth art board which reads type one. And that's gonna take you to the first type of tessellating pentagon, first discovered once again by Karl Reinhardt back in 1918. And you may recall from last week that these bold letters right here indicate the angles A, B, C, D and E, and then the italic letters represent the lengths of each one of the sides, also lettered a through e. And then we have a few rules down here. A plus B, these two angles right here, add up to 180 degrees, which means because all the angles in a pentagon add up to 540 degrees, that you take 540, subtract 180, and you get the fact that the three other angles add up to 360 degrees, which happens to be more incidental than meaningful. But the fact that A plus B equals 180 degrees is very meaningful because that means we're describing a half circle when we put these two angles right next to each other, and that allows us to repeat this pentagon. Notice that we also have a couple of sides right here, a and c, that are parallel to each other. They don't have to be the same length. But they do end up being parallel. So we're describing a kind of quadrangle, or four-sided shape. And so in the case of this specific example of a type-one pentagon, it repeats to create this simple pattern right here. Now I think it's pretty obvious how I assembled this pattern, but it's gonna become less obvious over time. So I wanna make things clear by turning on a couple more layers right here. And you can see that I've taken that original light blue shape, and then I rotated it 180 degrees, and moved it into place. And let me show you what that looks like just to make things perfectly clear. So anytime you see R in one of these slides, it indicates rotate, and where Illustrator is concerned, I prefer to rotate objects using the rotate tool. And then you could just alt or option click to bring up the rotate dialog box, enter 180 degrees, and click copy. And again just to make things clear, I'll go ahead and do it. I know this is pretty obvious. But again, it's gonna get less obvious over time. So I'll go ahead and turn that layer off. And then I'll grab this shape right there, and I'll make a copy of it by alt or option dragging it. And then I'll grab the rotate tool, which you can get by pressing the R key. I'll alt or option click down here, so below the shape in order to bring up the rotate dialog box. I'll change the angle value to 180 degrees, and then I'll just go ahead and click copy. And then I'll assign a different fill here inside the swatches panel. So notice that the fill is active. And I'll go ahead and select this color that I've created in advance called second for the second shape. And then I'll switch back to the black arrow tool by pressing the V key. And I'll go ahead and drag this guy by it's anchor point so that it snaps into place. And by the way, if you're seeing a bounding box on screen, you'll wanna go ahead and turn it off so that you can snap all the shapes into alignment, and you do that by going up to the view menu and choosing hide bounding box. If it's already off, the command will read show bounding box in which case, don't choose it, you don't need it. And I guess I went with a different color. But it doesn't really matter. Anyway, I'll go ahead and shift click on this guy, and then drag it by this anchor point like so, until it snaps into alignment. And then I'll press and hold the alt key, or the option key on the Mac, so I get that double arrowhead cursor. And then I'll release to create a copy. And then I would press control D or command D on the Mac a bunch of times in order to fill out that pattern. So that's one example. I'm just gonna go ahead and turn those layers off. Let's see another example of the type-one pentagon. Because I want you to understand, they don't have to be specific shapes. They just have to follow these rules right here. And so if I switch to the next art board by pressing shift page down, here we see a type-one variation in which not only do angles A and B add up to 180 degrees, but I've gone ahead and made the lengths of these two opposite sides, a and c, exactly the same, so that they'll exactly align to each other. And that's gonna end up looking like this. So slightly different. And you can see that we have an uninterrupted horizontal alignment. And this is just one way to arrange this tile by the way. So that's something to know as well. And I came with this arrangement like so. I went ahead and took that first tile, which I've colored a light orange, and I flipped it this time around vertically, which means that I flipped it across a horizontal axis. And just so that this file makes perfect sense, I'll go and turn on this demos layer so that you can see that flip indicates a reflection that is most easily pulled off using the reflect tool in my opinion, and in this case I went ahead and set the axis to horizontal and once again clicked copy. And to see what that looks like, I'll just go ahead and do it. I'll grab this guy, make a copy of him like so, and then switch to the reflect tool, which you can get by pressing the O key. And then I'll just alt or option click right there on one of the bottom anchor points. I'll set axis to horizontal so that we're flipping across the horizontal axis which strictly speaking is a vertical flip. And then I'll click the copy button in order to create a copy of that guy. Then I change his color here in the swatches panel. I press the V key to switch back to the black arrow tool. I shift click on this guy. And just drag the anchor point until it snaps into alignment, press and hold the alter option key in order to make a duplicate. And then press control D or command D on the Mac a bunch of times in order to fill out the pattern. Alright, I'm gonna go ahead and turn off these layers right here so that we can see the type-two pentagon, which looks more or less like this. The most important thing is that angles B and D add up to 180 degrees, and that way they can rest right next to each other and describe a half circle. So B fits into D, D's gonna fit into B. And then sides c and e are equal to each other, which is why I've colored them in red. And so if we repeat this tile, we'll end up with this pattern right here. So notice this time around, I've repeated the tile a total of four times in order to complete the pattern. And so I'll go ahead and turn on these two layers right here. So you can see I started with this light blue guy. I went ahead and flipped it vertically, so across the horizontal axis. And then I rotated it negative 20 degrees. And then I took both of the shapes and rotated them 180 degrees. And this is just one approach by the way, but it happens to work out very nicely. And so I'll just go ahead and grab this guy, make a copy of him just so we can see this happen. And by the way, you have to flip first and rotate second. If you rotate first, you would rotate the shape by positive 20 degrees instead of negative. But it makes the most sense to flip it first. So I'll go ahead and grab that reflect tool. I'll alt or option click somewhere up here above the shape to bring up the reflect dialog box. Axis is already set to horizontal so I'll just click copy in order to make a copy of that shape. I'll change its fill color to a darker shade of blue. And then I'll press and hold the control key or the command key on the Mac in order to get my black arrow tool on the fly. And I'll drag this anchor point to this location right there so it snaps into place. And then I'll switch to the rotate tool by pressing the R key. And then I'll alt or option click on that same anchor point to bring up the rotate dialog box. I'll change the angle value to negative 20, or if you prefer, you could cancel out here. And then I'll go up to the view menu and choose smart guides in order to turn the smart guides on. And with that target set to that anchor point, I'll just go ahead and drag this opposite anchor point until it snaps into alignment. And you can see in the heads-up display that I get a rotate value of negative 20 degrees. And then I would just go ahead and control shift click on this first shape, that's a command shift click on the Mac, in order to select it and I'll alt or option click above these shapes to bring up the rotate dialog box. I'll change the angle value to 180 degrees and click the copy button in order to create a copy of those shapes. I'll press the V key to switch back to my black arrow tool, and I'll drag this anchor point until it snaps into alignment like so. And then I might just go ahead and change the color of these shapes once again from the swatches panel. And I'll go ahead and shift click on the other shape so that they're all selected. And you would drag this anchor point right there until it snaps into alignment with this location. Press and hold the alt key or the option key on the Mac to make a copy and so forth. So for now, so for now we need to build these patterns manually. I'll show you a different way to build up the patterns in a future episode. Alright, I'm gonna go ahead and turn those layers off. And I'll press shift page down to advance to the next art board, so that we can see a representative of a type-three pentagon. And notice in this case that angles A, C and D, all three equal exactly 120 degrees, which means that B and E are left to add up to 180 degrees. That's not all that important. That's really incidental. But the fact that these three angles equal 120 degrees is very important, especially for angle A, because notice if you repeat it a total of three times, three times 120 gives you 360, which describes a circle. And we'll see what that looks like in just a moment. Meanwhile, sides a and b are equal to each other, whereas d is the sum of e and c. And so you can see I've gone ahead and colored side c orange, and side e green, so that it's obvious that they add up to d. And that's gonna become important as well. And so let's see what that looks like. This is the pattern, and notice that angle A is repeated against itself, as are sides a and b right there. And then here we have side d up toward the top with e and c repeated against it. Now this one is very easy to assemble. I'll just go ahead and show you that all you have to do is select that first shape right there, and then rotate it 120 degrees twice in a row, around this anchor point right there. Alright moving right along, let's check out a representative of a type-four pentagon, this guy right here. And notice that both angles B and D are 90 degrees, so we have right angles right here at this location, and this one as well. Sides b and c are equal as are sides d and e. And that way they'll work out when they're adjacent to each other, which looks like this. And so notice that we end up with these kind of stretch pentagons which I'm tracing right now. And so this is a representative of what's known as a Cairo tesselation, which is a specific variety of type four. And one way to put that together is to grab that first shape right there, rotate a copy of it 90 degrees so that this point right here which is angle C meets up with this point angle A. And then I took both of the shapes and rotated them 180 degrees. So again, that's one way to work. And that's just one example of a type-four pentagon. Here's another example right here. And so even though it looks quite a bit different than this one, it does obey the same rules. So angles B and D are both equal to 90 degrees. And that way B can repeat around itself as we'll see in just a moment. Sides b and c are the same length as are sides d and e. So b and d, they don't have to be anything like each other. It's just b equals c, d equals e. And that ends up looking like this. So we have this kind of arrangement of four propeller blades, if you will, or we have more of a stretch tile as I'm tracing right there. It depends how you want to think about it. And so I've put this together just using a straight rotation. So I took this first tile right there, and then I rotated a copy of it 90 degrees, a total of three times. Alright I'll go ahead and turn off those layers, and we'll take a look at the final Reinhardt pentagon which is type five, this guy right here. And in this case, angle A equals 60 degrees, exactly 60 degrees, which is 1/6 of circle. So we're gonna be able to put a bunch of angle As right next to each other. D equals 120 degrees, which is 1/3 of the circle as we're seeing right here. So we'll have some repetition there as well. And then we'll be able to put side a next to side b, and side d next to side e. And that's gonna look like this right here. And again, very easy to assemble. All you do is take that first shape and then you rotate a total of five copies 60 degrees. And everything fits together nicely. But the shape doesn't have to look exactly like this. You can stretch it out if you want to in order to create this type five variation. So again, A is still equal to 60 degrees. So same rules as ever. D equals 120, so we can repeat six As right next to each other and three Ds as we're seeing up here. Sides A and B are equal as are sides D and E, and that ends up looking like this. So again, we have propeller blades just as we did with the type four pentagon, but instead of seeing four blades, we're seeing a total of six blades. And you assemble that pattern in the exact same way. So you take that first blade right there, and you rotate five copies of it, exactly 60 degrees. And that friends is how you take advantage of the five Reinhardt pentagons first discovered as I record this, 100 years ago. Alright now if you're a member of lynda.com/linkedinlearning, I have a followup movie in which we take a look at the next three types of pentagons, types six, seven and eight right here, which were discovered by Richard Kirchner a full 50 years later in 1968, not to mention 50 years ago, and he even had the audacity to prove that these were the only eight tessellating irregular convex pentagons that were even possible. Obviously he was wrong as I'll prove next week when we take a look at what I'm calling the people's pentagons, which were discovered by folks just like you. Deke's Techniques, each and every week. Keep watching.

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