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

740 Types 6–8: The Kershner pentagons

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

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740 Types 6–8: The Kershner pentagons

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- [Instructor] In this movie, we'll take a look at what are known as Types 6-8 of the famous tessellating pentagons. And, so, if you're working in the same files I am, drop down to the bottom left corner of the screen. Click on this down pointing arrowhead, labeled Artboard Navigation. And, then, just go ahead and scroll down the menu until you see this guy, artboard 12 Type 6. And click on it, in order to navigate to that artboard. And, you'll see, whereas the Type 5 pentagon was first discovered by Karl Reinhardt, back in 1918, the Type 6 was discovered by a guy named Richard Kershner, a full 50 years later. And it ends up looking like this. So, recall that the bold letters indicate the angles between the sides. And the italic letters indicate the lengths of the sides themselves. And so the rules this time around are that angles B and D add up to 180 degrees. And, then, if you take angle B and multiply it times two, you get angle E, over here on the left hand side. Where the signs are concerned, B and C are equal to each other, as are sides A, D, and E. And that ends up repeating, like so. So that we have these tiles moving in opposite directions. Now, to assemble this guy, you take that first shape, which I filled with a light shade of blue, and then you rotate it around angle D, whatever it is. Because it doesn't necessarily have to be 130 degrees. But, in my case, it is. So you rotate it 130 degrees. And then you move it into place. Because here we have side E, and it ends up getting rotated into this position. So that it exactly neighbors side D. And, then, I took both of these shapes and rotated copies of them 180 degrees. And then you end up repeating the shapes so they fit next to each other. Alright, we'll go ahead and turn those layers off. And I'll press Shift + Page Down, to advance to the next artboard, which contains an example of the Type 7 pentagon. So, this time around, the rules are a little more obtuse. We're adding angle B to two times angle E, in order to get 360 degrees. And you can also multiply C times two and add it to D, to get that same amount. Also very important is the fact that all of the sides, except A, are exactly the same length. And so that ends up turning into this pattern, in which the original tile is repeated a total of eight times. Now this one's a little more gnarly than the stuff we've seen before. Notice, if I turn on this assembly layer right here, that I started with this light blue shape and then I went ahead and flipped it across the vertical axis. So, in other words, we have a horizontal flip. And, then, I rotate it by 69.87 degrees. And you may wonder, where in the world does that rotation come from. I haven't the vaguest idea. It just happens to work for this specific shape. And so, notice, if I go ahead and create a copy of this guy. And then I grab my Reflect tool, which you can get by pressing the O key. And I'll Alt or Option click on that anchor point right there, to bring up the Reflect dialogue box. I'll switch the axis to vertical and click Copy, in order to create a copy of that shape. And I'll change it to color, just so we match what's going on over here on the right. And then, I'll press the R key, to switch back to the Rotate tool. I'll go up to the View menu and, once again, turn on my Smart Guides. And then I'll click right there at that anchor point. And I'll drag this anchor point, like so, until it snaps into alignment. And you can see, in that heads up display, that I am rotating the shape by 69.87 degrees. And that just happens to work out. Next, I took that same light blue shape again. And I rotated it by -25.97 degrees, in order to create this purple-ish shape right here. And then I went ahead and rotated that purple shape in order to create this darker blue shape. And the angle value this time around is -154.02 degrees. Of course it is, why wouldn't that work? After that, I went ahead and grabbed all four of these shapes. And I flipped duplicates of them across the horizontal access. So, even though it doesn't look like it, I performed a vertical flip. And then I rotated all the shapes 43.91 degrees. And it ends up working out quite nicely. Now, of course, you don't need to color your shapes this way. I'm just doing so to make things as obvious as possible. But, of course, this color scheme ends up creating these kind of rivulets through the document. Alright, I'll go ahead and turn that pattern off. And then I'll press Shift + Page Down to advance to the third and final Kershner pentagon. Which is known as Type 8, and it looks like this. Once again, we have some fairly mysterious stuff going on where the angles are concerned. Two times B plus C, which happens to be 360 degrees. So that's different from what we had before. So notice we were still adding up to 360 degrees. But we were adding the angles differently. However, the sides add up just the same. So B, C, D, and E are all the same length. A is allowed to vary, and that ends up looking like this. So a little different than what we saw before. Here is the Type 7 pattern, and here is Type 8. Again, this is one of many possible variations. The tiles don't have to look exactly like this. As long as they follow the rules. And I assembled this guy a little more simply than the Type 7 pattern. I took this original light blue shape. And I rotated it -45.18 degrees. So, that part, I just had to feel my way through. And then I took both of those shapes and rotated copies of them by 180 degrees, in order to produce these next blue shapes. Then I took all four of them and I flipped them across the horizontal axis. So, in other words, I performed a vertical flip. Which is a little more obvious this time around. And then I rotated them -60.89 degrees, in order to line things up. And that takes care of the three Kershner pentagons. Which were first discovered 50 years after the Reinhardt pentagons. And, as I record this, 50 years ago.

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