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

738 Introducing the Pentagon Patterns

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

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738 Introducing the Pentagon Patterns

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- Hey gang, this is Deke McClelland, welcome to Deke's Techniques. Now this week marks the beginning of a deep dive that we're about to take, on a topic that I hope will interest those of you who are passionate about repeating tile patterns, i.e. tessellations. So, while it probably escaped your notice, this year marks the 100th anniversary of the discovery of, get this, the first tessellating irregular convex pentagon. And I know you're probably thinking, my word that is the best news I've ever heard, except for the part where I don't have the remotest idea of what you're talking about. So let me explain, if you will. We'll start with the three sided shape, aka triangle. Any triangle you draw with tessellate. So click three times with the pen tool, rotate the shape 180 degrees, and away you go. Then we have four sided shapes which are known as quadralaterals, they also tessellate. So this guy right here is an irregular one, but I've rotated it to four different angles and it tessellates beautifully, and this goes for parallelograms, as well as rectangles and, of course, squares. Then we have six sided shapes, i.e. hexagons. There are three types of hexagons that tessellate including this guy right here, which is type three by the way. They all include regular hexagons, which you already know tessellate because Illustrator and other programs offer hex patterns. What about shapes with seven, eight, or nine sides? Well if they're convex, and I'll show you what I mean by that in a moment, they cannot possibly tessellate, which leaves us with a missing shape, the five sided shape, or pentagon. Now the ancient Greeks proved that regular pentagons, like the Pentagon building in Washington, D.C. can not tessellate, but they suspected that there might be an irregular convex pentagon that could. They never found it, neither did the great pattern makers of Arabia. It wasn't until 1918 when a German mathematician named Karl Reinhardt discovered five types of tessellating pentagons, and when I say types I don't mean specific shapes, I mean classes of shapes. So for example, this is an example right here of type four but so this is, and they look very different than each other. Now over the years, more of these pentagons have been discovered and these days we know about 15 of them including this guy right here, type 15, which was discovered in the year 2015, get that. And notice, if you will, that it's the same pentagon rotated to 12 different angles. Now this might seem pretty complicated but I'm gonna make it absolute child's play starting inside this very movie. All right, let's start things off with a quick overview of the basic styles of five sided shapes, that is to say pentagons, that we have to work with. Starting with the regular pentagon which is the kind of pentagon you can create using the polygon tool here inside Illustrator. Now notice, where the diagram is concerned, that the angles between the sides appear in bold and then length of the sides appear in italics. And so for any regular polygon, each and every one of the angles equals 108 degrees, and if you take 108 and multiply it times five then you end up with 540 degrees which is the total angle value for any five sided shape. Meanwhile, where a regular pentagon is concerned, the length of each and every side is identical. Now something else I want you to note, as least when working in Illustrator, is that the program measures pentagons and other regular polygons, and so notice if I select the shape, that I've gone ahead and identified the center, and so if I were to grab the line segment tool and then I need to make sure that my smart guides are turned on so I can see the heads up display, I'll go ahead and drag from that center to any one of the anchor points along the perimeter of the shape, and you can see that my distance value right there is 170 points, which tells me the length of that line. And so it doesn't matter which one of these anchor points I drag to, I'm gonna get that same value. And so I'll just go ahead and get rid of that line and then I'll select the polygon tool, from the shape tool fly out menu, and I'll go ahead and click on that center point in order to bring up the polygon dialogue box, I'll change the number of sides to five, and then I'll shift tab to the radius value, and I'll take it up to 170 points just as I saw a moment ago, and I'll click okay, and I end up with that exact same shape. And here inside the Illustrator CC, I can click on the word shape up here in the horizontal control panel, at which point I can change the number of sides if I want to, and I can change that radius value, and Illustrator does a good job of showing you what the radius looks like, at least where a hexagon is concerned. In any event, the problem where a regular pentagon is concerned, is that it doesn't tessellate properly. And so notice, if I turn on this layer right here, and I'll go ahead and deselect the artwork as well, that I've done my best to create a repeating pattern using that regular pentagon, but it just so happens it's impossible to tile a regular pentagon monohedrally, this is to say with a single tile, at least at 2D Euclidean space, in other spaces you can do some wacky stuff, but where two dimensional design is concerned, 108 degrees doesn't divide evenly into a 360 degree circle and so we end up with these diamond shaped gaps in between. All right, I'll go ahead and turn off that layer and I will switch to the next artboard so that we can see a second style of pentagon which is an irregular non-convex shape. And so what we have here is a group of five different angles, identified in bold once again, which add up to 540 degrees, however, they can all be different, and you can have one indent, which I've identified here in red, or it is possible to have a pentagon with two indents but I've never seen one tessellate. Meanwhile, your sides can be any length whatsoever. And so if we were to take this shape, the one we're currently seeing, and repeat it by rotating, flipping, and moving it around, then you can see that it tessellates wonderfully. The problem is, no one's really classified how many varieties of these irregular non-convex repeating pentagons that there are. And so while it's very cool, this is not something that we're gonna delve into in this particular technique. Instead, we're gonna take a look at irregular convex pentagons which look like this guy right here. And so notice that once again, all of our angles add up to 540 degrees, we have no indents, hence the word convex up here, and once again our sides can be any length. And so over the last 100 years, and to this very date, 15 types of irregular convex pentagons are known to tile monohedrally including the one that we're seeing here, which is known as a type five pentagon, and the reason they're called types is because they're not specific shapes, in other words, a type five pentagon doesn't have to look exactly like this guy here, rather the varieties of shapes as we'll see. And that's my overview of the basic varieties of pentagons that you can't and sometimes can tessellate monohedrally, that is by taking a single shape and rotating, flipping, and/or moving it, not only here inside Illustrator, but in 2D space. All right, so are you excited? I hope so, because next week we're gonna begin our look at every single one of those 15 types starting with the first five types from Karl Reinhardt. Deke's Techniques each and every week, you've gotta keep watching.

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