My first patterns were created while I was a teenager in the 1960's, long before I had a computer, or even seen one. I called my first patterns "Inventions" or "My Invention". Over a three year period I invented five patterns. While I explored them a bit at Thomas More School (a boarding high school in Harrisville, New Hampshire), the task of growing them on large taped-together sheets of graph paper was done at home. I devoted sparse attention to them while at school. My primary hobby at that time was weather observing and forecasting. Many of the illustrations shown here are from later computerized renderings of the patterns.
This section describes the five original patterns in detail.
Before discovering the patterns I used to draw things like this:
Figure 1.1 - Reconstructed childhood drawings
I was into "iterative" drawings. In the first example, a tree stump was drawn with progressively smaller cut-off branches. In the second example, a star was drawn inside a pentagon, and then progressively smaller stars within the pentagons formed by the stars. In the third example, two smaller crosses were formed by proportionately smaller lines, the process may be repeated.
My breakthrough was to develop another method of pattern creation using the repetitive use of lines which remained the same size!
The first Invention was the Line Pattern. I was fascinated by it. I discovered the pattern while drawing iterations of letters of the alphabet while I was a freshman in high school. After I got home for the summer, I bought some graph paper with five grid units per inch and hand-drew the pattern over several days while keeping track with a graph of how many lines were drawn in each round. I then filled in the pattern's shapes using colored pencils. All shapes with an equal number of grid squares inside their perimeters were assigned and drawn a single color. After I was done, I taped the drawing onto a wall. The Line Pattern is incredibly simple to draw. Here is how to draw it:
Obtain some graph paper and start at a single point, as shown at the far left of Figure 1.2. From the starting point, draw a line to the next grid intersection to the left of the starting point and draw a line to the next grid intersection to the right. You now have drawn a Module which in this case consists of two branches, one to the left and one to the right, each one unit long. The resulting pattern drawn so far appears as a single line two units long. Of course, when I hand-drew the Line Pattern, I drew the first line with a single stroke two units long instead of two strokes one unit long, although this Module's formal definition consists of the two branches each one unit long.
From each end of the line just drawn, draw an additional line, growing one unit to the left and one unit to the right using the frame of reference you would have if you were facing the directions, in turn, that the ends of the previously drawn line were growing toward. This completes the second Cycle. A Cycle consists of drawing all of the Modules in a pattern's pattern definition in Module rounds, in the sequence of their definition. The Line Pattern has only one Module in its pattern definition, so the act of drawing the first Module completed the Line Pattern's first Cycle.
In the third Cycle, the Module is drawn four times and something important happens. There are two collisions each involving two Modules. In this case, we will obey a frequently used collision rule, which says, "If two end points collide, then stop growing."
In the sixth Cycle, there is a new kind of collision. Some of the branches collide with the previously grown structure. The collision rule for this circumstance says, "If an end point collides with the previously grown structure, then stop growing." The previously grown structure in this case consists of lines drawn in prior Cycles. Also, all along we have been following a yet unstated rule which says, "If an end point does not collide with anything, then keep growing."
The eighth Cycle of the Line Pattern may be drawn with just two lines, each eight units long.
The Line Pattern is closely tied in with the "powers of two" sequence. The outer shape of the growing pattern returns to form an approximate square when the pattern is grown a Power-Cycle, that is, a number of Cycles in the 1, 2, 4, 8, 16, 32, 64, 128... sequence. Two raised to the seventh power is equal to 2 x 2 x 2 x 2 x 2 x 2 x 2 or 128. Seven Power-Cycles is equal to 128 Cycles. The "zeroth" Power-Cycle is two raised to the zeroth power or one Cycle. Figure 1.3 shows the Line Pattern after it has grown 128 Cycles and Figure 1.4 shows the Line Pattern after it has grown 496 Cycles (at a reduced scale). After 512 Cycles, the Line Pattern will once again form a near square.
The second Invention was the V Pattern. I believe I invented this pattern about a year after the Line Pattern - the spring of 1966 is my best guess. Many beautiful things happen in this pattern. I hand-drew this pattern to 64 Cycles at the same scale as the original Line Pattern.
The V Pattern, like the Line Pattern, has only one Module with two branches and is easy to draw, although a few new wrinkles are introduced. The branches, instead of going to the left and to the right, go diagonally forward left and diagonally forward right. Figure 1.5a is a picture of the Module for which the V Pattern was named.
I have a standard way of illustrating the way Modules are defined. The Modules are always shown in their "upright position", that is, the forward relative direction of the Module points to the "straight up" or "north" direction. The forward relative direction for any Module is the direction that the branch, from which the Module will grow, was growing.
The V Pattern introduces the importance of the way patterns start. In Figure 1.6a and 1.6b, the V Pattern is shown at eight Cycles. In Figure 1.6a, the first branches were grown in the horizontal and vertical directions. In Figure 1.6b, the first branches were diagonal. Notice how both expressions of the V Pattern contain the identical number of grown branches and resulting enclosed shapes. The only difference is in the way the shapes interlock. The first V Pattern that I drew started as the one shown in Figure 1.6a.
An expression of the V Pattern is illustrated in Figure 1.7. The basic octagon shape may be seen as eight triangular wedges. Within the triangles are smaller triangles. This is one of the signature features of Expansions.
The third Invention was the Great Pattern (also referred to in private communications as "Book 3"). There were two major breakthroughs in this pattern. This was the first pattern to have more than one Module in a Cycle (it has two, Figure 1.8a) and it was the first pattern to have a Module with more than one part. I think I invented this pattern sometime in the first half of 1967. The Great Pattern is one of my favorites when it comes to ranking them by the enclosed shapes produced. I hand-drew and colored this pattern to 31-1/2 Cycles.
The Great Pattern introduces a new problem: which Module to draw first. Figure 1.8b shows the first few Cycles of the Great Pattern with diagonal starting directions and the "staple"-shaped Module drawn before the "V" Module. Some of the Modules are shown in Figure 1.8b with thinner lines to increase the clarity of the growing process. The "staple" Module has two first-part branches and each first-part branch has one second-part branch. The point at which the first-part ends and the second-part begins is called a part point. The ends of the second-parts in the "staple" Module and ends of the first-parts of the "V" Module are all referred to as end points.
The Great Pattern introduces some of the most complicated collisions rules that I have devised for Expansions. There are three new rules obeyed by the Great Pattern. The first new rule is, "If a part point collides with an end point generated in the previous Module round, then keep growing." The second new rule is, "If a first-part of a Module grows in the same space as another first-part growing in the same or opposite direction, then keep growing." The third rule is, "If two part points collide, then keep growing."
The rule which says to stop growing when end points collide was bypassed by a bypass function in the first Cycle of the Great Pattern as shown in Figure 1.8b. More about this later.
Figures 1.9a and 1.9b show two expressions of the Great Pattern. In Figure 1.9a, the "staple" Module was drawn first. In Figure 1.9b, the "V" Module was drawn first. Some of the pattern's enclosed shapes have been filled with black to show the signature triangular pattern of repetition frequently found. In Figure 1.9b, as can be seen, the choice of start-up settings does not result in a true expression of the pattern. A determination of a pattern's trueness may be seen by growing a pattern for additional rounds. Figure 1.9c shows a signature shape emerging (arrow) when the expression of the Great Pattern in Figure 1.9b is grown another Power-Cycle. Another clue to the problem is that Figure 1.9a was grown four complete Cycles before it appeared to complete a Power-Cycle, while Figure 1.9b was grown only 3.5 Cycle Figure 1.9a shows a true expression of the Great Pattern. When I first drew the Great Pattern by hand, I drew the expression in 1.9b / 1.9c. This expression of the Great Pattern also does not require the suspension of any collision rules in the first Cycle. Years passed before I understood the problem I had with the Great Pattern.
The fourth Invention was the Egg Pattern (also referred to in private communications as "Book 4" as it was the fourth pattern recorded in a notebook). The "Egg" name comes from four egg-like enclosed shapes this pattern makes after about 18 Cycles. Drawing the pattern by hand and seeing the Eggs appear for the first time was fantastic! This pattern is simpler to draw than the Great Pattern. It has three Modules (Figure 1.10a). This is the first pattern that contained a Module with a long branch. Figure 1.10b shows the first two Cycles of the Egg Pattern.
I think I invented this pattern in the second half of 1967. I hand-drew and colored it to about 17 Cycles, just enough to enclose the four eggs (Figure 1.11). I no longer have the original hand drawn image, but I have do have a color photograph of it, the paper looks like it had been taped to my bedroom wall for some time. The photograph has a "1968" film processing date on it.
The first Module has long branches. The point between the starting point of the long branch and the end point of the branch is called a unit point. Unit points are so called because the sections of branch between the starting point and the unit point and the unit point and the end point (or part point if there should be a second-part) are called units. (See Appendix A for an illustration of a generic Module). Within a given pattern, all units of all Modules have to be drawn with the same length. The length of the unit is called the pattern's scale and may be set to any number of grid units.
Unit points introduce some more rules. The first new rule is, "If a unit point collides with an end point generated in the previous Module round, then stop growing." The second new rule is, "If two unit points collide, then stop growing." (A complete list of collision combinations and default settings for the "grow" or "stop" outcomes are listed in Appendix A).
Like the V Pattern and the Great Pattern, the Egg Pattern grows into somewhat of an octagon. However, unlike the other two patterns, a pattern of repetition only appears for the four wide sides of the octagon, while non-repetition appears to be the rule for the four narrow, diagonal sides. In Figure 1.11, the enclosed shapes filled with black form the standard triangular repetition pattern that characterizes many patterns. The gray signature shapes of the narrow sides do not repeat with regularity. I found this randomness baffling and wanted to know more about it! I got a lot of grid paper and taped the sheets together to explore one of the random sections. I still have this exploration in a box somewhere. After I computerized Expansions I later grew the Egg Pattern well over 1000 cycles and the narrow sides do not repeat! Note how signature components of the narrow and wide octagon sides crossbreed in each other's domain as the pattern unfolds. Crossbreeding is a standard feature of many patterns.
Figure 1.11b - One quadrant of the Egg Pattern after more the 300 cycles
The "octant" section doesn't repeat, appearing random
Note: Image compressed 2:1 and then inverted
The Eggs pattern using a general hand-colored scheme based somewhat from the original drawing done in 1967 (see below)
The narrow "octant" shows some repetition early on, but grows increasingly random
Original Egg Pattern hand draw on taped together sheets of graph paper - created in 1967
Same image, increased contrast
The fifth and last Invention of the first stage was named "Square 2" because it formed a square similar to the Line Pattern, except that in this case all four sides were equal in length. I drew and colored this pattern to 32 Cycles. I believe I first drew this pattern in 1968, probably in the summer. The Modules and first two Cycles of the Square 2 Pattern are shown in Figures 1.12a and 1.12b. There is question as to whether the pattern shown here is identical to my original fifth pattern. If not, the original pattern was very similar to what is shown here.
Figure 1.13a shows the repetitive tiling characteristic of the "Square 2" pattern. When this pattern is shown at a small scale, Figure 1.13b, the overall expression is similar to the expression of the Line Pattern in Figure 1.4. One thing that fascinates me about both the Line Pattern and Square 2 is the effect that reminds me so much of ruler markers. A steel ruler my father had in a workroom had long markers for the inches, shorter markers for the 1/2 inch interval, yet shorter markers for the 1/4 and 3/4 inch intervals, and so on down to 1/64th of an inch. I remember looking at the Line Pattern, thinking about how it reminded me so much of a ruler's pattern of subdivision.
Animation Supplement - Restored June 2002
Select an image for animation.
Note: Some animation jumps more than one cycle at a time
All animations end up with a little bit of Color Expansions activity
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© John S. Stokes III - Inventor, Artist, Puzzle Maker & Webmaster