Add some more to the generalization sections.

content/blog/modulo_patterns/index.md

@@ -277,37 +277,136 @@ factor with 9; 3, 6, 9, 12, and so on. However, those can't all serve as cycle l
 cycles can't get longer than 9. This leaves us with 3, 6, and 9 as _possible_ cycle lengths,
 none of which are divisible by 4. We've eliminated the possibility of spirals!
 
-{{< todo >}}
-This doesn't get to the bottom of it all.
-{{< /todo >}}
-
 ### Generalizing to Arbitrary Divisors
 The trick was easily executable on paper because there's an easy way to compute the remainder of a number
 when dividing by 9 (adding up the digits). However, we have a computer, and we don't need to fall back on such
 cool-but-complicated techniques. To replicate our original behavior, we can just write:
 
-```
+```Ruby
 def sum_digits(n)
   x = n % 9
   x == 0 ? 9 : x
 end
 ```
 
-But now, we can change the `9` to something else. Any number we pick, so long as it isn't
+But now, we can change the `9` to something else. There are some numbers we'd like to avoid - specifically,
-{{< sidenote "right" "div-4-note" "divisible by 4," >}}
+we want to avoid those numbers that would allow for cycles of length 4 (or of a length divisible by 4).
-"Wait", you might be thinking, "I thought you said that 4 can't have a common factor with the divisor,
+If we didn't avoid them, we might run into infinite loops, where our pencil might end up moving
-and that means any even numbers are out, too."<br>
+further and further from the center.
-<br>
+
-Good observation. Although the path-not-divisible-by-four condition is certainly sufficient, it is not
+Actually, let's revisit that. When we were playing with paths of length \\(k\\) while dividing by 9,
-<em>necessary</em>. There seems to be another, less restrictive, condition at play here: even numbers work fine. I haven't
+we noted that the only _possible_ values of \\(k\\) are those that share a common factor with 9,
-figured out what it is, but we might as well make use of it.
+specifically 3, 6 and 9. But that's not quite as strong as it could be: try as you might, but
-{{< /sidenote >}} will work. I'll pick primes for good measure. Here are a few good ones from using 13
+you will not find a cycle of length 6 when dividing by 9. The same is true if we pick 6 instead of 9,
+and try to find a cycle of length 4. Even though 4 _does_ have a common factor with 6, and thus
+is not ruled out as a valid cycle by our previous condition, we don't find any cycles of length 4.
+
+So what is it that _really_ determines if there can be cycles or not?
+
+Let's do some more playing around. What _are_ the actual cycle lengths when we divide by 9?
+For all but two numbers, the cycle lengths are 9. The two special numbers are 6 and 3, and they end up
+with a cycle length of 3. From this, we can say that the cycle length seems to depend on whether or
+nor our \\(n\\) has any common factors with the divisor.
+
+Let's explore this some more with a different divisor, say 12. We fill find that 8 has a cycle length
+of 3, 7 has a cycle length of 12, 9 has a cycle length of 4. What's
+happening here? To see, let's divide 12 __by these cycle lengths__. For 8, we get (12/3) = 4.
+For 7, this works out to 1. For 9, it works out to 3. These new numbers, 4, 1, and 3, are actually
+the __greatest common factors__ of 8, 3, and 7 with 12, respectively. The greatest common factor
+of two numbers is the largest number that divides them both. We thus write down our guess
+for the length of a cycle:
+
+{{< latex >}}
+k = \frac{d}{\text{gcd}(d,n)}
+{{< /latex >}}
+
+Where \\(d\\) is our divisor, which has been 9 until just recently. Indeed, this equation is in agreement
+with our experiment for \\(d = 9\\), too. Why might this be? Let's start once again with
+our equation for paths of length \\(k\\):
+
+{{< latex >}}
+kn \equiv 0\ (\text{mod}\ d)
+{{< /latex >}}
+
+Here we've replaced 9 with \\(d\\), since we're trying to make it work for _any_ divisor, not just 9.
+Now, suppose the greatest common divisor of \\(n\\) and \\(d\\) is some number \\(f\\). Then,
+since this number divides \\(n\\) and \\(d\\), we can write \\(n=fm\\) for some \\(m\\), and
+\\(d=fg\\) for some \\(g\\). We can rewrite our equation as follows:
+
+{{< latex >}}
+kfm \equiv 0\ (\text{mod}\ fg)
+{{< /latex >}}
+
+We can simplify this a little bit. Recall that what this equation _really_ means is that the
+difference of \\(kfm\\) and \\(0\\), which is just \\(kfm\\) is divisible by \\(fg\\):
+
+{{< latex >}}
+fg|kfm
+{{< /latex >}}
+
+But if \\(fg\\) divides \\(kfm\\), it must be that \\(g\\) divides \\(km\\)! This, in turn, means
+we can write:
+
+{{< latex >}}
+g|km
+{{< /latex >}}
+
+We also know that \\(g\\) and \\(m\\) have no common factors -- they were all divided out from \\(d\\)
+and \\(n\\) when we divided by \\(f\\)! We can thus further simplify our claim:
+
+{{< latex >}}
+g|k
+{{< /latex >}}
+
+This says that \\(k\\) must be divisible by \\(g\\). Recall that we got \\(g\\) by dividing
+\\(d\\) by \\(f\\), which is our largest common factor -- aka \\(\\text{gcd}(d,n)\\). We can thus
+write:
+
+{{< latex >}}
+\frac{d}{\text{gcd}(d,n)}|k
+{{< /latex >}}
+
+All that's left is to pick the smallest \\(k\\) that fits this description (that would be the first
+point at which our sequence of multiples of \\(n\\) will loop). Zero is divisible by anything, but
+alas, our sequence numbers start at 1 -- zero's out. The next best thing is \\(d/\\text{gcd}(d,n)\\).
+And there we have it: the first point at which our sequence loops, just like we guessed.
+
+Lastly, recall that a cycle occurs whenever a \\(k\\) is a multiple of 4. Now that we know what
+\\(k\\) is, we can restate this as "\\(d/\\text{gcd}(d,n)\\) is divisible by 4". But if we pick
+\\(n=d-1\\), the
+{{< sidenote "right" "coprime-note" "greatest common factor has to be \(1\)," >}}
+Wait, why is <em>this</em> true? Well, suppose some number \(f\) divides both \(d\) and \(d-1\).
+In that case, we can write \(d=af\), and \((d-1)=bf\). Subtracting one equation from the other:
+{{< latex >}}
+1 = (a-b)f
+{{< /latex >}}
+But this means that 1 is divisible by \(f\)! That's only possible if \(f=1\). Thus, the only
+number that divides \(x\) and \(x-1\) is 1; that's our greatest common factor.
+{{< /sidenote >}} which means that our condition further simplifies to
+"\\(d\\) is divisible by 4".
+Thus, we can state simply that any divisor divisible by 4 is off-limits, as it will induce loops.
+For example, pick \\(d=4\\). Running our algorithm for \\(n=d-1=3\\), we indeed find an infinite
+spiral:
+
+{{< figure src="pattern_3_4.svg" caption="Spiral generated by the number 3 with divisor 4." class="tiny" alt="Spiral generated by the number 3 by summing digits." >}}
+
+Let's try again. Pick \\(d=8\\); then, for \\(n=d-1=7\\), we also get a spiral:
+
+{{< figure src="pattern_7_8.svg" caption="Spiral generated by the number 7 with divisor 8." class="tiny" alt="Spiral generated by the number 7 by summing digits." >}}
+
+A poem comes to mind:
+> Turning and turning in the wydening gyre
+>
+> The falcon cannot hear the falconner;
+
+Fortunately, there are plenty of numbers that are not divisible by four, and we can pick
+any of them! I'll pick primes for good measure. Here are a few good ones from using 13
 (which corresponds to summing digits of base-14 numbers):
 
 {{< figure src="pattern_8_13.svg" caption="Pattern generated by the number 8 in base 14." class="tiny" alt="Pattern generated by the number 8 by summing digits." >}}
 {{< figure src="pattern_4_13.svg" caption="Pattern generated by the number 4 in base 14." class="tiny" alt="Pattern generated by the number 4 by summing digits." >}}
 
-Here are a few from dividing by 17 (base-18 numbers).
+Here's one from dividing by 17 (base-18 numbers).
 
 {{< figure src="pattern_5_17.svg" caption="Pattern generated by the number 5 in base 18." class="tiny" alt="Pattern generated by the number 5 by summing digits." >}}
 
@@ -316,3 +415,86 @@ Finally, base-30:
 {{< figure src="pattern_2_29.svg" caption="Pattern generated by the number 2 in base 30." class="tiny" alt="Pattern generated by the number 2 by summing digits." >}}
 
 {{< figure src="pattern_6_29.svg" caption="Pattern generated by the number 6 in base 30." class="tiny" alt="Pattern generated by the number 6 by summing digits." >}}
+
+### Generalizing to Arbitrary Numbers of Directions
+What if we didn't turn 90 degrees each time? What, if, instead, we turned 120 degrees (so that
+turning 3 times, not 4, would leave you facing the same direction you started)? We can pretty easily
+do that, too. Let's call this number of turns \\(c\\). Up until now, we had \\(c=4\\).
+
+First, let's update our condition. Before, we had "\\(d\\) cannot be divisible by 4". Now,
+we aren't constraining ourselves to only 4, but rather using a generic variable \\(c\\).
+We then end up with "\\(d\\) cannot be divisible by \\(c\\)". For instance, suppose we kept
+our divisor as 9 for the time being, but started turning 3 times instead of 4. This
+violates are divisibility condtion, and we once again end up with a spiral:
+
+{{< figure src="pattern_8_9_t3.svg" caption="Pattern generated by the number 8 in base 10 while turning 3 times." class="tiny" alt="Pattern generated by the number 3 by summing digits and turning 120 degrees." >}}
+
+If, on the other hand, we pick \\(d=8\\) and \\(c=3\\), we get patterns for all numbers just like we hoped.
+Here's one such pattern:
+
+{{< figure src="pattern_7_8_t3.svg" caption="Pattern generated by the number 7 in base 9 while turning 3 times." class="tiny" alt="Pattern generated by the number 7 by summing digits in base 9 and turning 120 degrees." >}}
+
+Hold on a moment; it's actully not so obvious why our condition _still_ works. When we just turned
+on a grid, things were simple. As long as we didn't end up facing the same way we started, we will
+eventually perform the exact same motions in reverse. The same is not true when turning 120 degrees, like
+we suggested. Here's a circle with the turn angles labeled:
+
+{{< figure src="turn_3_1.png" caption="Orientations when turning 120 degrees" class="small" alt="Possible orientations when turning 120 degrees." >}}
+
+We never quite do the exact _opposite_ of any one of our movements. So then, will we come back to the
+origin anyway? Well, let's start simple. Suppose we always turn by exactly one 120-degree increment
+(we might end up turning more or less, just like we may end up turning left, right, or back in the
+90 degree case). Now,
+
+1. Suppose that having performed one complete cycle, we end up away from the center
+by \\(dx\\) on the \\(x\\)-axis, and \\(dy\\) on the \\(y\\)-axis (we do this without loss
+of generality).
+2. We are now turned around by 120 degrees, so once we perform the cycle again, we end up offset
+by \\(dx(\\cos 120)-dy(\\sin 120)\\) on the \\(x\\)-axis, and \\(dx(\\sin 120)+dy(\\cos 120)\\) on
+the \\(y\\)-axis (I got this from the [rotation matrx](https://en.wikipedia.org/wiki/Rotation_matrix)
+page on Wikipedia).
+3. After one more step, we end up with having rotated a total of 240 degrees. As we perform the cycle
+again, we end up having moved by an additional \\(dx(\\cos 240)-dy(\\sin 240)\\) and \\(dx(\\sin 240)+dy(\\cos 240)\\).
+
+Summing up all of these changes, we get:
+
+{{< latex >}}
+    dx(\cos0+\cos120+\cos240) + dy(\sin0+\sin120+\sin240)
+{{< /latex >}}
+
+Why don't we start trying to write this in terms of variables already? For some number of turns
+\\(c\\), a single turn is \\(360/c\\) degrees. We start having turned 0 degrees, then progress
+to having turned \\(360/c\\) degrees, then \\(2\times360/c\\), and so on until \\((c-1)360/c\\).
+We can write this using _summation notation_ (and radians instead of degrees) as follows:
+
+{{< latex >}}
+    \begin{aligned}
+        x &= dx\left[\sum_{i=0}^{c-1} \cos\left(i\frac{2\pi}{c}\right)\right] -
+        dy\left[\sum_{i=0}^{c-1} \sin\left(i\frac{2\pi}{c}\right)\right] \\
+        y &= dx\left[\sum_{i=0}^{c-1} \sin\left(i\frac{2\pi}{c}\right)\right] +
+        dy\left[\sum_{i=0}^{c-1} \cos\left(i\frac{2\pi}{c}\right)\right] \\
+    \end{aligned}
+{{< /latex >}}
+
+For reasons beyond the scope of this article, sums like those between the square brackets
+in the above equations _always_ equal zero. This means that after all the turns have been made,
+we get \\(x=0\\) and \\(y=0\\) -- back at the origin, where we started!
+
+{{< todo >}}Maybe we can prove the sin/cos thing? {{< /todo >}}
+
+What if we turn by 240 degrees at a time: 2 turns instead of 1? Even though we first turn
+a whole 240 degrees, the second time we turn we "overshoot" our initial bearing, and end up at 120 degrees
+compared to it. As soon as we turn 240 more degrees (turning the third time), we end up back at 0.
+In short, even though we "visited" each bearing in a different order, we visited them all.
+
+{{< figure src="turn_3_2.png" caption="Orientations when turning 120 degrees, twice at a time" class="small" alt="Possible orientations when turning 120 degrees, twice at a time." >}}
+
+Let's try put some mathematical backing to this "visited them all" idea. 
+
+{{< todo >}}Remainders, visited them all, etc.{{< /todo >}}
+
+But let's not be so boring. We can branch out some, of course.
+
+{{< figure src="pattern_1_7_t5.svg" caption="Pattern generated by the number 1 in base 8 while turning 5 times." class="tiny" alt="Pattern generated by the number 1 by summing digits in base 8 and turning 72 degrees." >}}
+
+{{< figure src="pattern_3_11_t6.svg" caption="Pattern generated by the number 3 in base 12 while turning 6 times." class="tiny" alt="Pattern generated by the number 3 by summing digits in base 12 and turning 60 degrees." >}}

content/blog/modulo_patterns/pattern_1_7_t5.svg

@@ -0,0 +1,80 @@
+<?xml version="1.0" standalone="no"?>
+<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
+
+<svg width="311.2461179749811" height="301.4817981115687" 
+  xmlns="http://www.w3.org/2000/svg"
+  xmlns:xlink="http://www.w3.org/1999/xlink">
+
+
+<line x1="47.08203932499384" y1="78.81909602355867" x2="187.08203932499384" y2="78.81909602355867" style="stroke:black; stroke-width:5"/>
+<circle cx="187.08203932499384" cy="78.81909602355867" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="187.08203932499384" y1="78.81909602355867" x2="193.26237921249282" y2="97.84022634946174" style="stroke:black; stroke-width:5"/>
+<circle cx="193.26237921249282" cy="97.84022634946174" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="193.26237921249282" y1="97.84022634946174" x2="160.9016994374949" y2="121.35163644116068" style="stroke:black; stroke-width:5"/>
+<circle cx="160.9016994374949" cy="121.35163644116068" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="160.9016994374949" y1="121.35163644116068" x2="112.36067977499803" y2="86.08452130361229" style="stroke:black; stroke-width:5"/>
+<circle cx="112.36067977499803" cy="86.08452130361229" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="112.36067977499803" y1="86.08452130361229" x2="137.0820393249938" y2="10.0" style="stroke:black; stroke-width:5"/>
+<circle cx="137.0820393249938" cy="10.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="137.0820393249938" y1="10.0" x2="237.0820393249938" y2="10.0" style="stroke:black; stroke-width:5"/>
+<circle cx="237.0820393249938" cy="10.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="237.0820393249938" y1="10.0" x2="274.1640786499875" y2="124.12678195541842" style="stroke:black; stroke-width:5"/>
+<circle cx="274.1640786499875" cy="124.12678195541842" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="274.1640786499875" y1="124.12678195541842" x2="160.90169943749487" y2="206.41671727636466" style="stroke:black; stroke-width:5"/>
+<circle cx="160.90169943749487" cy="206.41671727636466" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="160.90169943749487" y1="206.41671727636466" x2="144.72135954999595" y2="194.6610122305152" style="stroke:black; stroke-width:5"/>
+<circle cx="144.72135954999595" cy="194.6610122305152" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="144.72135954999595" y1="194.6610122305152" x2="157.0820393249938" y2="156.61875157870907" style="stroke:black; stroke-width:5"/>
+<circle cx="157.0820393249938" cy="156.61875157870907" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="157.0820393249938" y1="156.61875157870907" x2="217.0820393249938" y2="156.61875157870907" style="stroke:black; stroke-width:5"/>
+<circle cx="217.0820393249938" cy="156.61875157870907" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="217.0820393249938" y1="156.61875157870907" x2="241.8033988749896" y2="232.70327288232136" style="stroke:black; stroke-width:5"/>
+<circle cx="241.8033988749896" cy="232.70327288232136" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="241.8033988749896" y1="232.70327288232136" x2="160.90169943749487" y2="291.4817981115687" style="stroke:black; stroke-width:5"/>
+<circle cx="160.90169943749487" cy="291.4817981115687" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="160.90169943749487" y1="291.4817981115687" x2="63.81966011250117" y2="220.94756783647193" style="stroke:black; stroke-width:5"/>
+<circle cx="63.81966011250117" cy="220.94756783647193" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="63.81966011250117" y1="220.94756783647193" x2="107.08203932499376" y2="87.7996555551504" style="stroke:black; stroke-width:5"/>
+<circle cx="107.08203932499376" cy="87.7996555551504" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="107.08203932499376" y1="87.7996555551504" x2="127.08203932499376" y2="87.7996555551504" style="stroke:black; stroke-width:5"/>
+<circle cx="127.08203932499376" cy="87.7996555551504" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="127.08203932499376" y1="87.7996555551504" x2="139.44271909999168" y2="125.84191620695655" style="stroke:black; stroke-width:5"/>
+<circle cx="139.44271909999168" cy="125.84191620695655" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="139.44271909999168" y1="125.84191620695655" x2="90.90169943749484" y2="161.10903134450496" style="stroke:black; stroke-width:5"/>
+<circle cx="90.90169943749484" cy="161.10903134450496" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="90.90169943749484" y1="161.10903134450496" x2="26.18033988749903" y2="114.0862111611071" style="stroke:black; stroke-width:5"/>
+<circle cx="26.18033988749903" cy="114.0862111611071" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="26.18033988749903" y1="114.0862111611071" x2="57.082039324993744" y2="18.98055953159174" style="stroke:black; stroke-width:5"/>
+<circle cx="57.082039324993744" cy="18.98055953159174" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="57.082039324993744" y1="18.98055953159174" x2="177.08203932499376" y2="18.98055953159174" style="stroke:black; stroke-width:5"/>
+<circle cx="177.08203932499376" cy="18.98055953159174" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="177.08203932499376" y1="18.98055953159174" x2="220.34441853748638" y2="152.12847181291323" style="stroke:black; stroke-width:5"/>
+<circle cx="220.34441853748638" cy="152.12847181291323" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="220.34441853748638" y1="152.12847181291323" x2="204.16407864998746" y2="163.88417685876271" style="stroke:black; stroke-width:5"/>
+<circle cx="204.16407864998746" cy="163.88417685876271" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="204.16407864998746" y1="163.88417685876271" x2="171.80339887498957" y2="140.37276676706375" style="stroke:black; stroke-width:5"/>
+<circle cx="171.80339887498957" cy="140.37276676706375" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="171.80339887498957" y1="140.37276676706375" x2="190.34441853748638" y2="83.30937578935456" style="stroke:black; stroke-width:5"/>
+<circle cx="190.34441853748638" cy="83.30937578935456" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="190.34441853748638" y1="83.30937578935456" x2="270.3444185374864" y2="83.30937578935456" style="stroke:black; stroke-width:5"/>
+<circle cx="270.3444185374864" cy="83.30937578935456" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="270.3444185374864" y1="83.30937578935456" x2="301.2461179749811" y2="178.4150274188699" style="stroke:black; stroke-width:5"/>
+<circle cx="301.2461179749811" cy="178.4150274188699" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="301.2461179749811" y1="178.4150274188699" x2="204.16407864998743" y2="248.94925769396673" style="stroke:black; stroke-width:5"/>
+<circle cx="204.16407864998743" cy="248.94925769396673" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="204.16407864998743" y1="248.94925769396673" x2="90.90169943749476" y2="166.65932237302047" style="stroke:black; stroke-width:5"/>
+<circle cx="90.90169943749476" cy="166.65932237302047" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="90.90169943749476" y1="166.65932237302047" x2="97.0820393249937" y2="147.6381920471174" style="stroke:black; stroke-width:5"/>
+<circle cx="97.0820393249937" cy="147.6381920471174" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="97.0820393249937" y1="147.6381920471174" x2="137.0820393249937" y2="147.6381920471174" style="stroke:black; stroke-width:5"/>
+<circle cx="137.0820393249937" cy="147.6381920471174" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="137.0820393249937" y1="147.6381920471174" x2="155.62305898749057" y2="204.70158302482662" style="stroke:black; stroke-width:5"/>
+<circle cx="155.62305898749057" cy="204.70158302482662" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="155.62305898749057" y1="204.70158302482662" x2="90.90169943749476" y2="251.7244032082245" style="stroke:black; stroke-width:5"/>
+<circle cx="90.90169943749476" cy="251.7244032082245" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="90.90169943749476" y1="251.7244032082245" x2="10.0" y2="192.9458779789772" style="stroke:black; stroke-width:5"/>
+<circle cx="10.0" cy="192.9458779789772" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="10.0" y1="192.9458779789772" x2="47.082039324993666" y2="78.81909602355874" style="stroke:black; stroke-width:5"/>
+<circle cx="47.082039324993666" cy="78.81909602355874" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+
+</svg>

content/blog/modulo_patterns/pattern_3_11_t6.svg

@@ -0,0 +1,142 @@
+<?xml version="1.0" standalone="no"?>
+<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
+
+<svg width="600.0000000000006" height="574.2562584220407" 
+  xmlns="http://www.w3.org/2000/svg"
+  xmlns:xlink="http://www.w3.org/1999/xlink">
+
+
+<line x1="140.00000000000088" y1="287.12812921102045" x2="360.00000000000085" y2="287.12812921102045" style="stroke:black; stroke-width:5"/>
+<circle cx="360.00000000000085" cy="287.12812921102045" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="360.00000000000085" y1="287.12812921102045" x2="390.00000000000085" y2="339.08965343808677" style="stroke:black; stroke-width:5"/>
+<circle cx="390.00000000000085" cy="339.08965343808677" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="390.00000000000085" y1="339.08965343808677" x2="330.0000000000009" y2="443.0127018922194" style="stroke:black; stroke-width:5"/>
+<circle cx="330.0000000000009" cy="443.0127018922194" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="330.0000000000009" y1="443.0127018922194" x2="150.0000000000009" y2="443.0127018922194" style="stroke:black; stroke-width:5"/>
+<circle cx="150.0000000000009" cy="443.0127018922194" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="150.0000000000009" y1="443.0127018922194" x2="140.00000000000088" y2="425.6921938165306" style="stroke:black; stroke-width:5"/>
+<circle cx="140.00000000000088" cy="425.6921938165306" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="140.00000000000088" y1="425.6921938165306" x2="180.00000000000085" y2="356.4101615137755" style="stroke:black; stroke-width:5"/>
+<circle cx="180.00000000000085" cy="356.4101615137755" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="180.00000000000085" y1="356.4101615137755" x2="320.00000000000085" y2="356.4101615137755" style="stroke:black; stroke-width:5"/>
+<circle cx="320.00000000000085" cy="356.4101615137755" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="320.00000000000085" y1="356.4101615137755" x2="420.00000000000085" y2="529.6152422706632" style="stroke:black; stroke-width:5"/>
+<circle cx="420.00000000000085" cy="529.6152422706632" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="420.00000000000085" y1="529.6152422706632" x2="400.00000000000085" y2="564.2562584220407" style="stroke:black; stroke-width:5"/>
+<circle cx="400.00000000000085" cy="564.2562584220407" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="400.00000000000085" y1="564.2562584220407" x2="300.00000000000085" y2="564.2562584220407" style="stroke:black; stroke-width:5"/>
+<circle cx="300.00000000000085" cy="564.2562584220407" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="300.00000000000085" y1="564.2562584220407" x2="220.0000000000008" y2="425.69219381653056" style="stroke:black; stroke-width:5"/>
+<circle cx="220.0000000000008" cy="425.69219381653056" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="220.0000000000008" y1="425.69219381653056" x2="330.0000000000006" y2="235.166604983954" style="stroke:black; stroke-width:5"/>
+<circle cx="330.0000000000006" cy="235.166604983954" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="330.0000000000006" y1="235.166604983954" x2="390.0000000000006" y2="235.166604983954" style="stroke:black; stroke-width:5"/>
+<circle cx="390.0000000000006" cy="235.166604983954" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="390.0000000000006" y1="235.166604983954" x2="450.0000000000006" y2="339.0896534380866" style="stroke:black; stroke-width:5"/>
+<circle cx="450.0000000000006" cy="339.0896534380866" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="450.0000000000006" y1="339.0896534380866" x2="360.0000000000007" y2="494.97422611928556" style="stroke:black; stroke-width:5"/>
+<circle cx="360.0000000000007" cy="494.97422611928556" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="360.0000000000007" y1="494.97422611928556" x2="340.0000000000007" y2="494.97422611928556" style="stroke:black; stroke-width:5"/>
+<circle cx="340.0000000000007" cy="494.97422611928556" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="340.0000000000007" y1="494.97422611928556" x2="300.0000000000006" y2="425.69219381653056" style="stroke:black; stroke-width:5"/>
+<circle cx="300.0000000000006" cy="425.69219381653056" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="300.0000000000006" y1="425.69219381653056" x2="370.00000000000057" y2="304.4486372867091" style="stroke:black; stroke-width:5"/>
+<circle cx="370.00000000000057" cy="304.4486372867091" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="370.00000000000057" y1="304.4486372867091" x2="570.0000000000006" y2="304.4486372867091" style="stroke:black; stroke-width:5"/>
+<circle cx="570.0000000000006" cy="304.4486372867091" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="570.0000000000006" y1="304.4486372867091" x2="590.0000000000006" y2="339.0896534380866" style="stroke:black; stroke-width:5"/>
+<circle cx="590.0000000000006" cy="339.0896534380866" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="590.0000000000006" y1="339.0896534380866" x2="540.0000000000006" y2="425.6921938165305" style="stroke:black; stroke-width:5"/>
+<circle cx="540.0000000000006" cy="425.6921938165305" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="540.0000000000006" y1="425.6921938165305" x2="380.00000000000057" y2="425.6921938165305" style="stroke:black; stroke-width:5"/>
+<circle cx="380.00000000000057" cy="425.6921938165305" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="380.00000000000057" y1="425.6921938165305" x2="270.0000000000005" y2="235.16660498395402" style="stroke:black; stroke-width:5"/>
+<circle cx="270.0000000000005" cy="235.16660498395402" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="270.0000000000005" y1="235.16660498395402" x2="300.00000000000045" y2="183.2050807568877" style="stroke:black; stroke-width:5"/>
+<circle cx="300.00000000000045" cy="183.2050807568877" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="300.00000000000045" y1="183.2050807568877" x2="420.00000000000045" y2="183.2050807568877" style="stroke:black; stroke-width:5"/>
+<circle cx="420.00000000000045" cy="183.2050807568877" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="420.00000000000045" y1="183.2050807568877" x2="510.00000000000045" y2="339.08965343808666" style="stroke:black; stroke-width:5"/>
+<circle cx="510.00000000000045" cy="339.08965343808666" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="510.00000000000045" y1="339.08965343808666" x2="500.00000000000045" y2="356.4101615137754" style="stroke:black; stroke-width:5"/>
+<circle cx="500.00000000000045" cy="356.4101615137754" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="500.00000000000045" y1="356.4101615137754" x2="420.00000000000045" y2="356.4101615137754" style="stroke:black; stroke-width:5"/>
+<circle cx="420.00000000000045" cy="356.4101615137754" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="420.00000000000045" y1="356.4101615137754" x2="350.00000000000034" y2="235.16660498395407" style="stroke:black; stroke-width:5"/>
+<circle cx="350.00000000000034" cy="235.16660498395407" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="350.00000000000034" y1="235.16660498395407" x2="450.0000000000002" y2="61.96152422706625" style="stroke:black; stroke-width:5"/>
+<circle cx="450.0000000000002" cy="61.96152422706625" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="450.0000000000002" y1="61.96152422706625" x2="490.0000000000002" y2="61.96152422706625" style="stroke:black; stroke-width:5"/>
+<circle cx="490.0000000000002" cy="61.96152422706625" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="490.0000000000002" y1="61.96152422706625" x2="540.0000000000002" y2="148.5640646055101" style="stroke:black; stroke-width:5"/>
+<circle cx="540.0000000000002" cy="148.5640646055101" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="540.0000000000002" y1="148.5640646055101" x2="460.0000000000003" y2="287.1281292110203" style="stroke:black; stroke-width:5"/>
+<circle cx="460.0000000000003" cy="287.1281292110203" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="460.0000000000003" y1="287.1281292110203" x2="240.00000000000028" y2="287.12812921102034" style="stroke:black; stroke-width:5"/>
+<circle cx="240.00000000000028" cy="287.12812921102034" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="240.00000000000028" y1="287.12812921102034" x2="210.00000000000026" y2="235.16660498395402" style="stroke:black; stroke-width:5"/>
+<circle cx="210.00000000000026" cy="235.16660498395402" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="210.00000000000026" y1="235.16660498395402" x2="270.0000000000001" y2="131.24355652982135" style="stroke:black; stroke-width:5"/>
+<circle cx="270.0000000000001" cy="131.24355652982135" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="270.0000000000001" y1="131.24355652982135" x2="450.0000000000001" y2="131.24355652982135" style="stroke:black; stroke-width:5"/>
+<circle cx="450.0000000000001" cy="131.24355652982135" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="450.0000000000001" y1="131.24355652982135" x2="460.0000000000001" y2="148.5640646055101" style="stroke:black; stroke-width:5"/>
+<circle cx="460.0000000000001" cy="148.5640646055101" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="460.0000000000001" y1="148.5640646055101" x2="420.0000000000001" y2="217.84609690826522" style="stroke:black; stroke-width:5"/>
+<circle cx="420.0000000000001" cy="217.84609690826522" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="420.0000000000001" y1="217.84609690826522" x2="280.0000000000001" y2="217.84609690826525" style="stroke:black; stroke-width:5"/>
+<circle cx="280.0000000000001" cy="217.84609690826525" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="280.0000000000001" y1="217.84609690826525" x2="180.00000000000006" y2="44.64101615137757" style="stroke:black; stroke-width:5"/>
+<circle cx="180.00000000000006" cy="44.64101615137757" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="180.00000000000006" y1="44.64101615137757" x2="200.00000000000003" y2="10.0" style="stroke:black; stroke-width:5"/>
+<circle cx="200.00000000000003" cy="10.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="200.00000000000003" y1="10.0" x2="300.00000000000006" y2="10.0" style="stroke:black; stroke-width:5"/>
+<circle cx="300.00000000000006" cy="10.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="300.00000000000006" y1="10.0" x2="380.00000000000006" y2="148.56406460551017" style="stroke:black; stroke-width:5"/>
+<circle cx="380.00000000000006" cy="148.56406460551017" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="380.00000000000006" y1="148.56406460551017" x2="270.0000000000001" y2="339.08965343808677" style="stroke:black; stroke-width:5"/>
+<circle cx="270.0000000000001" cy="339.08965343808677" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="270.0000000000001" y1="339.08965343808677" x2="210.00000000000014" y2="339.08965343808677" style="stroke:black; stroke-width:5"/>
+<circle cx="210.00000000000014" cy="339.08965343808677" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="210.00000000000014" y1="339.08965343808677" x2="150.00000000000009" y2="235.1666049839541" style="stroke:black; stroke-width:5"/>
+<circle cx="150.00000000000009" cy="235.1666049839541" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="150.00000000000009" y1="235.1666049839541" x2="239.99999999999997" y2="79.28203230275507" style="stroke:black; stroke-width:5"/>
+<circle cx="239.99999999999997" cy="79.28203230275507" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="239.99999999999997" y1="79.28203230275507" x2="260.0" y2="79.28203230275507" style="stroke:black; stroke-width:5"/>
+<circle cx="260.0" cy="79.28203230275507" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="260.0" y1="79.28203230275507" x2="300.0" y2="148.56406460551017" style="stroke:black; stroke-width:5"/>
+<circle cx="300.0" cy="148.56406460551017" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="300.0" y1="148.56406460551017" x2="230.0" y2="269.8076211353316" style="stroke:black; stroke-width:5"/>
+<circle cx="230.0" cy="269.8076211353316" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="230.0" y1="269.8076211353316" x2="30.000000000000018" y2="269.8076211353316" style="stroke:black; stroke-width:5"/>
+<circle cx="30.000000000000018" cy="269.8076211353316" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="30.000000000000018" y1="269.8076211353316" x2="10.0" y2="235.16660498395407" style="stroke:black; stroke-width:5"/>
+<circle cx="10.0" cy="235.16660498395407" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="10.0" y1="235.16660498395407" x2="59.99999999999993" y2="148.56406460551017" style="stroke:black; stroke-width:5"/>
+<circle cx="59.99999999999993" cy="148.56406460551017" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="59.99999999999993" y1="148.56406460551017" x2="219.99999999999994" y2="148.56406460551017" style="stroke:black; stroke-width:5"/>
+<circle cx="219.99999999999994" cy="148.56406460551017" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="219.99999999999994" y1="148.56406460551017" x2="329.99999999999994" y2="339.08965343808666" style="stroke:black; stroke-width:5"/>
+<circle cx="329.99999999999994" cy="339.08965343808666" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="329.99999999999994" y1="339.08965343808666" x2="299.99999999999994" y2="391.051177665153" style="stroke:black; stroke-width:5"/>
+<circle cx="299.99999999999994" cy="391.051177665153" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="299.99999999999994" y1="391.051177665153" x2="179.99999999999997" y2="391.051177665153" style="stroke:black; stroke-width:5"/>
+<circle cx="179.99999999999997" cy="391.051177665153" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="179.99999999999997" y1="391.051177665153" x2="89.99999999999989" y2="235.1666049839541" style="stroke:black; stroke-width:5"/>
+<circle cx="89.99999999999989" cy="235.1666049839541" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="89.99999999999989" y1="235.1666049839541" x2="99.99999999999987" y2="217.8460969082653" style="stroke:black; stroke-width:5"/>
+<circle cx="99.99999999999987" cy="217.8460969082653" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="99.99999999999987" y1="217.8460969082653" x2="179.99999999999986" y2="217.8460969082653" style="stroke:black; stroke-width:5"/>
+<circle cx="179.99999999999986" cy="217.8460969082653" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="179.99999999999986" y1="217.8460969082653" x2="249.9999999999999" y2="339.08965343808677" style="stroke:black; stroke-width:5"/>
+<circle cx="249.9999999999999" cy="339.08965343808677" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="249.9999999999999" y1="339.08965343808677" x2="149.99999999999994" y2="512.2947341949745" style="stroke:black; stroke-width:5"/>
+<circle cx="149.99999999999994" cy="512.2947341949745" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="149.99999999999994" y1="512.2947341949745" x2="109.99999999999993" y2="512.2947341949745" style="stroke:black; stroke-width:5"/>
+<circle cx="109.99999999999993" cy="512.2947341949745" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="109.99999999999993" y1="512.2947341949745" x2="59.99999999999987" y2="425.6921938165306" style="stroke:black; stroke-width:5"/>
+<circle cx="59.99999999999987" cy="425.6921938165306" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="59.99999999999987" y1="425.6921938165306" x2="139.99999999999977" y2="287.12812921102034" style="stroke:black; stroke-width:5"/>
+<circle cx="139.99999999999977" cy="287.12812921102034" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+
+</svg>

content/blog/modulo_patterns/pattern_3_4.svg

@@ -0,0 +1,42 @@
+<?xml version="1.0" standalone="no"?>
+<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
+
+<svg width="220.0" height="200.0" 
+  xmlns="http://www.w3.org/2000/svg"
+  xmlns:xlink="http://www.w3.org/1999/xlink">
+
+
+<line x1="10" y1="10" x2="90.0" y2="10.0" style="stroke:black; stroke-width:5"/>
+<circle cx="90.0" cy="10.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="90.0" y1="10.0" x2="90.0" y2="70.0" style="stroke:black; stroke-width:5"/>
+<circle cx="90.0" cy="70.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="90.0" y1="70.0" x2="50.0" y2="70.0" style="stroke:black; stroke-width:5"/>
+<circle cx="50.0" cy="70.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="50.0" y1="70.0" x2="49.99999999999999" y2="50.00000000000001" style="stroke:black; stroke-width:5"/>
+<circle cx="49.99999999999999" cy="50.00000000000001" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="49.99999999999999" y1="50.00000000000001" x2="130.0" y2="50.00000000000001" style="stroke:black; stroke-width:5"/>
+<circle cx="130.0" cy="50.00000000000001" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="130.0" y1="50.00000000000001" x2="130.0" y2="110.0" style="stroke:black; stroke-width:5"/>
+<circle cx="130.0" cy="110.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="130.0" y1="110.0" x2="90.0" y2="110.0" style="stroke:black; stroke-width:5"/>
+<circle cx="90.0" cy="110.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="90.0" y1="110.0" x2="90.0" y2="90.0" style="stroke:black; stroke-width:5"/>
+<circle cx="90.0" cy="90.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="90.0" y1="90.0" x2="170.0" y2="90.0" style="stroke:black; stroke-width:5"/>
+<circle cx="170.0" cy="90.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="170.0" y1="90.0" x2="170.0" y2="150.0" style="stroke:black; stroke-width:5"/>
+<circle cx="170.0" cy="150.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="170.0" y1="150.0" x2="130.0" y2="150.0" style="stroke:black; stroke-width:5"/>
+<circle cx="130.0" cy="150.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="130.0" y1="150.0" x2="130.0" y2="130.0" style="stroke:black; stroke-width:5"/>
+<circle cx="130.0" cy="130.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="130.0" y1="130.0" x2="210.0" y2="130.0" style="stroke:black; stroke-width:5"/>
+<circle cx="210.0" cy="130.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="210.0" y1="130.0" x2="210.0" y2="190.0" style="stroke:black; stroke-width:5"/>
+<circle cx="210.0" cy="190.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="210.0" y1="190.0" x2="170.0" y2="190.0" style="stroke:black; stroke-width:5"/>
+<circle cx="170.0" cy="190.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="170.0" y1="190.0" x2="170.0" y2="170.0" style="stroke:black; stroke-width:5"/>
+<circle cx="170.0" cy="170.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+
+</svg>

content/blog/modulo_patterns/pattern_7_8.svg

@@ -0,0 +1,74 @@
+<?xml version="1.0" standalone="no"?>
+<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
+
+<svg width="420.0" height="400.0" 
+  xmlns="http://www.w3.org/2000/svg"
+  xmlns:xlink="http://www.w3.org/1999/xlink">
+
+
+<line x1="10" y1="10" x2="170.0" y2="10.0" style="stroke:black; stroke-width:5"/>
+<circle cx="170.0" cy="10.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="170.0" y1="10.0" x2="170.0" y2="150.0" style="stroke:black; stroke-width:5"/>
+<circle cx="170.0" cy="150.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="170.0" y1="150.0" x2="50.0" y2="150.00000000000003" style="stroke:black; stroke-width:5"/>
+<circle cx="50.0" cy="150.00000000000003" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="50.0" y1="150.00000000000003" x2="49.999999999999986" y2="50.000000000000014" style="stroke:black; stroke-width:5"/>
+<circle cx="49.999999999999986" cy="50.000000000000014" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="49.999999999999986" y1="50.000000000000014" x2="130.0" y2="50.000000000000014" style="stroke:black; stroke-width:5"/>
+<circle cx="130.0" cy="50.000000000000014" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="130.0" y1="50.000000000000014" x2="130.0" y2="110.00000000000001" style="stroke:black; stroke-width:5"/>
+<circle cx="130.0" cy="110.00000000000001" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="130.0" y1="110.00000000000001" x2="89.99999999999999" y2="110.00000000000001" style="stroke:black; stroke-width:5"/>
+<circle cx="89.99999999999999" cy="110.00000000000001" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="89.99999999999999" y1="110.00000000000001" x2="89.99999999999999" y2="90.00000000000001" style="stroke:black; stroke-width:5"/>
+<circle cx="89.99999999999999" cy="90.00000000000001" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="89.99999999999999" y1="90.00000000000001" x2="250.0" y2="90.00000000000001" style="stroke:black; stroke-width:5"/>
+<circle cx="250.0" cy="90.00000000000001" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="250.0" y1="90.00000000000001" x2="250.0" y2="230.0" style="stroke:black; stroke-width:5"/>
+<circle cx="250.0" cy="230.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="250.0" y1="230.0" x2="130.0" y2="230.0" style="stroke:black; stroke-width:5"/>
+<circle cx="130.0" cy="230.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="130.0" y1="230.0" x2="130.0" y2="130.0" style="stroke:black; stroke-width:5"/>
+<circle cx="130.0" cy="130.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="130.0" y1="130.0" x2="210.0" y2="130.0" style="stroke:black; stroke-width:5"/>
+<circle cx="210.0" cy="130.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="210.0" y1="130.0" x2="210.0" y2="190.0" style="stroke:black; stroke-width:5"/>
+<circle cx="210.0" cy="190.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="210.0" y1="190.0" x2="170.0" y2="190.0" style="stroke:black; stroke-width:5"/>
+<circle cx="170.0" cy="190.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="170.0" y1="190.0" x2="170.0" y2="170.0" style="stroke:black; stroke-width:5"/>
+<circle cx="170.0" cy="170.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="170.0" y1="170.0" x2="330.0" y2="170.0" style="stroke:black; stroke-width:5"/>
+<circle cx="330.0" cy="170.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="330.0" y1="170.0" x2="330.0" y2="310.0" style="stroke:black; stroke-width:5"/>
+<circle cx="330.0" cy="310.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="330.0" y1="310.0" x2="210.0" y2="310.0" style="stroke:black; stroke-width:5"/>
+<circle cx="210.0" cy="310.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="210.0" y1="310.0" x2="209.99999999999997" y2="210.0" style="stroke:black; stroke-width:5"/>
+<circle cx="209.99999999999997" cy="210.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="209.99999999999997" y1="210.0" x2="289.99999999999994" y2="210.0" style="stroke:black; stroke-width:5"/>
+<circle cx="289.99999999999994" cy="210.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="289.99999999999994" y1="210.0" x2="289.99999999999994" y2="270.0" style="stroke:black; stroke-width:5"/>
+<circle cx="289.99999999999994" cy="270.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="289.99999999999994" y1="270.0" x2="249.99999999999997" y2="270.0" style="stroke:black; stroke-width:5"/>
+<circle cx="249.99999999999997" cy="270.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="249.99999999999997" y1="270.0" x2="249.99999999999997" y2="250.0" style="stroke:black; stroke-width:5"/>
+<circle cx="249.99999999999997" cy="250.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="249.99999999999997" y1="250.0" x2="410.0" y2="250.0" style="stroke:black; stroke-width:5"/>
+<circle cx="410.0" cy="250.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="410.0" y1="250.0" x2="410.0" y2="390.0" style="stroke:black; stroke-width:5"/>
+<circle cx="410.0" cy="390.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="410.0" y1="390.0" x2="290.0" y2="390.0" style="stroke:black; stroke-width:5"/>
+<circle cx="290.0" cy="390.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="290.0" y1="390.0" x2="289.99999999999994" y2="290.0" style="stroke:black; stroke-width:5"/>
+<circle cx="289.99999999999994" cy="290.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="289.99999999999994" y1="290.0" x2="370.0" y2="290.0" style="stroke:black; stroke-width:5"/>
+<circle cx="370.0" cy="290.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="370.0" y1="290.0" x2="370.0" y2="350.0" style="stroke:black; stroke-width:5"/>
+<circle cx="370.0" cy="350.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="370.0" y1="350.0" x2="330.0" y2="350.0" style="stroke:black; stroke-width:5"/>
+<circle cx="330.0" cy="350.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="330.0" y1="350.0" x2="330.0" y2="330.0" style="stroke:black; stroke-width:5"/>
+<circle cx="330.0" cy="330.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+
+</svg>

content/blog/modulo_patterns/pattern_7_8_t3.svg

@@ -0,0 +1,58 @@
+<?xml version="1.0" standalone="no"?>
+<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
+
+<svg width="240.00000000000014" height="227.84609690826517" 
+  xmlns="http://www.w3.org/2000/svg"
+  xmlns:xlink="http://www.w3.org/1999/xlink">
+
+
+<line x1="70.00000000000014" y1="96.60254037844375" x2="230.00000000000014" y2="96.60254037844375" style="stroke:black; stroke-width:5"/>
+<circle cx="230.00000000000014" cy="96.60254037844375" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="230.00000000000014" y1="96.60254037844375" x2="160.00000000000017" y2="217.84609690826517" style="stroke:black; stroke-width:5"/>
+<circle cx="160.00000000000017" cy="217.84609690826517" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="160.00000000000017" y1="217.84609690826517" x2="100.00000000000011" y2="113.92304845413257" style="stroke:black; stroke-width:5"/>
+<circle cx="100.00000000000011" cy="113.92304845413257" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="100.00000000000011" y1="113.92304845413257" x2="200.0000000000001" y2="113.92304845413257" style="stroke:black; stroke-width:5"/>
+<circle cx="200.0000000000001" cy="113.92304845413257" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="200.0000000000001" y1="113.92304845413257" x2="160.00000000000014" y2="183.20508075688767" style="stroke:black; stroke-width:5"/>
+<circle cx="160.00000000000014" cy="183.20508075688767" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="160.00000000000014" y1="183.20508075688767" x2="130.0000000000001" y2="131.24355652982138" style="stroke:black; stroke-width:5"/>
+<circle cx="130.0000000000001" cy="131.24355652982138" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="130.0000000000001" y1="131.24355652982138" x2="170.0000000000001" y2="131.24355652982138" style="stroke:black; stroke-width:5"/>
+<circle cx="170.0000000000001" cy="131.24355652982138" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="170.0000000000001" y1="131.24355652982138" x2="160.0000000000001" y2="148.56406460551014" style="stroke:black; stroke-width:5"/>
+<circle cx="160.0000000000001" cy="148.56406460551014" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="160.0000000000001" y1="148.56406460551014" x2="80.00000000000003" y2="10.0" style="stroke:black; stroke-width:5"/>
+<circle cx="80.00000000000003" cy="10.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="80.00000000000003" y1="10.0" x2="220.00000000000003" y2="10.0" style="stroke:black; stroke-width:5"/>
+<circle cx="220.00000000000003" cy="10.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="220.00000000000003" y1="10.0" x2="160.00000000000006" y2="113.92304845413264" style="stroke:black; stroke-width:5"/>
+<circle cx="160.00000000000006" cy="113.92304845413264" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="160.00000000000006" y1="113.92304845413264" x2="110.00000000000001" y2="27.320508075688803" style="stroke:black; stroke-width:5"/>
+<circle cx="110.00000000000001" cy="27.320508075688803" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="110.00000000000001" y1="27.320508075688803" x2="190.00000000000003" y2="27.320508075688803" style="stroke:black; stroke-width:5"/>
+<circle cx="190.00000000000003" cy="27.320508075688803" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="190.00000000000003" y1="27.320508075688803" x2="160.00000000000003" y2="79.28203230275513" style="stroke:black; stroke-width:5"/>
+<circle cx="160.00000000000003" cy="79.28203230275513" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="160.00000000000003" y1="79.28203230275513" x2="140.00000000000003" y2="44.641016151377585" style="stroke:black; stroke-width:5"/>
+<circle cx="140.00000000000003" cy="44.641016151377585" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="140.00000000000003" y1="44.641016151377585" x2="160.00000000000003" y2="44.641016151377585" style="stroke:black; stroke-width:5"/>
+<circle cx="160.00000000000003" cy="44.641016151377585" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="160.00000000000003" y1="44.641016151377585" x2="80.00000000000006" y2="183.20508075688778" style="stroke:black; stroke-width:5"/>
+<circle cx="80.00000000000006" cy="183.20508075688778" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="80.00000000000006" y1="183.20508075688778" x2="10.0" y2="61.961524227066406" style="stroke:black; stroke-width:5"/>
+<circle cx="10.0" cy="61.961524227066406" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="10.0" y1="61.961524227066406" x2="130.0" y2="61.961524227066406" style="stroke:black; stroke-width:5"/>
+<circle cx="130.0" cy="61.961524227066406" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="130.0" y1="61.961524227066406" x2="80.00000000000001" y2="148.56406460551028" style="stroke:black; stroke-width:5"/>
+<circle cx="80.00000000000001" cy="148.56406460551028" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="80.00000000000001" y1="148.56406460551028" x2="39.999999999999986" y2="79.28203230275521" style="stroke:black; stroke-width:5"/>
+<circle cx="39.999999999999986" cy="79.28203230275521" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="39.999999999999986" y1="79.28203230275521" x2="99.99999999999999" y2="79.28203230275521" style="stroke:black; stroke-width:5"/>
+<circle cx="99.99999999999999" cy="79.28203230275521" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="99.99999999999999" y1="79.28203230275521" x2="79.99999999999999" y2="113.92304845413277" style="stroke:black; stroke-width:5"/>
+<circle cx="79.99999999999999" cy="113.92304845413277" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="79.99999999999999" y1="113.92304845413277" x2="69.99999999999999" y2="96.60254037844399" style="stroke:black; stroke-width:5"/>
+<circle cx="69.99999999999999" cy="96.60254037844399" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+
+</svg>

content/blog/modulo_patterns/pattern_8_9_t3.svg

@@ -0,0 +1,64 @@
+<?xml version="1.0" standalone="no"?>
+<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
+
+<svg width="379.9999999999999" height="262.487113059643" 
+  xmlns="http://www.w3.org/2000/svg"
+  xmlns:xlink="http://www.w3.org/1999/xlink">
+
+
+<line x1="10" y1="10" x2="190.0" y2="10.0" style="stroke:black; stroke-width:5"/>
+<circle cx="190.0" cy="10.0" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="190.0" y1="10.0" x2="110.00000000000003" y2="148.5640646055102" style="stroke:black; stroke-width:5"/>
+<circle cx="110.00000000000003" cy="148.5640646055102" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="110.00000000000003" y1="148.5640646055102" x2="39.99999999999997" y2="27.32050807568882" style="stroke:black; stroke-width:5"/>
+<circle cx="39.99999999999997" cy="27.32050807568882" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="39.99999999999997" y1="27.32050807568882" x2="159.99999999999997" y2="27.32050807568882" style="stroke:black; stroke-width:5"/>
+<circle cx="159.99999999999997" cy="27.32050807568882" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="159.99999999999997" y1="27.32050807568882" x2="109.99999999999999" y2="113.9230484541327" style="stroke:black; stroke-width:5"/>
+<circle cx="109.99999999999999" cy="113.9230484541327" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="109.99999999999999" y1="113.9230484541327" x2="69.99999999999994" y2="44.64101615137763" style="stroke:black; stroke-width:5"/>
+<circle cx="69.99999999999994" cy="44.64101615137763" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="69.99999999999994" y1="44.64101615137763" x2="129.99999999999994" y2="44.64101615137763" style="stroke:black; stroke-width:5"/>
+<circle cx="129.99999999999994" cy="44.64101615137763" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="129.99999999999994" y1="44.64101615137763" x2="109.99999999999997" y2="79.28203230275517" style="stroke:black; stroke-width:5"/>
+<circle cx="109.99999999999997" cy="79.28203230275517" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="109.99999999999997" y1="79.28203230275517" x2="99.99999999999997" y2="61.961524227066406" style="stroke:black; stroke-width:5"/>
+<circle cx="99.99999999999997" cy="61.961524227066406" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="99.99999999999997" y1="61.961524227066406" x2="279.99999999999994" y2="61.961524227066406" style="stroke:black; stroke-width:5"/>
+<circle cx="279.99999999999994" cy="61.961524227066406" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="279.99999999999994" y1="61.961524227066406" x2="200.0" y2="200.5255888325766" style="stroke:black; stroke-width:5"/>
+<circle cx="200.0" cy="200.5255888325766" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="200.0" y1="200.5255888325766" x2="129.99999999999994" y2="79.28203230275523" style="stroke:black; stroke-width:5"/>
+<circle cx="129.99999999999994" cy="79.28203230275523" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="129.99999999999994" y1="79.28203230275523" x2="249.99999999999994" y2="79.28203230275523" style="stroke:black; stroke-width:5"/>
+<circle cx="249.99999999999994" cy="79.28203230275523" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="249.99999999999994" y1="79.28203230275523" x2="199.99999999999994" y2="165.8845726811991" style="stroke:black; stroke-width:5"/>
+<circle cx="199.99999999999994" cy="165.8845726811991" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="199.99999999999994" y1="165.8845726811991" x2="159.9999999999999" y2="96.60254037844403" style="stroke:black; stroke-width:5"/>
+<circle cx="159.9999999999999" cy="96.60254037844403" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="159.9999999999999" y1="96.60254037844403" x2="219.9999999999999" y2="96.60254037844403" style="stroke:black; stroke-width:5"/>
+<circle cx="219.9999999999999" cy="96.60254037844403" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="219.9999999999999" y1="96.60254037844403" x2="199.9999999999999" y2="131.2435565298216" style="stroke:black; stroke-width:5"/>
+<circle cx="199.9999999999999" cy="131.2435565298216" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="199.9999999999999" y1="131.2435565298216" x2="189.9999999999999" y2="113.92304845413281" style="stroke:black; stroke-width:5"/>
+<circle cx="189.9999999999999" cy="113.92304845413281" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="189.9999999999999" y1="113.92304845413281" x2="369.9999999999999" y2="113.92304845413281" style="stroke:black; stroke-width:5"/>
+<circle cx="369.9999999999999" cy="113.92304845413281" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="369.9999999999999" y1="113.92304845413281" x2="289.9999999999999" y2="252.48711305964298" style="stroke:black; stroke-width:5"/>
+<circle cx="289.9999999999999" cy="252.48711305964298" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="289.9999999999999" y1="252.48711305964298" x2="219.99999999999983" y2="131.2435565298216" style="stroke:black; stroke-width:5"/>
+<circle cx="219.99999999999983" cy="131.2435565298216" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="219.99999999999983" y1="131.2435565298216" x2="339.9999999999999" y2="131.2435565298216" style="stroke:black; stroke-width:5"/>
+<circle cx="339.9999999999999" cy="131.2435565298216" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="339.9999999999999" y1="131.2435565298216" x2="289.9999999999999" y2="217.8460969082655" style="stroke:black; stroke-width:5"/>
+<circle cx="289.9999999999999" cy="217.8460969082655" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="289.9999999999999" y1="217.8460969082655" x2="249.99999999999983" y2="148.56406460551042" style="stroke:black; stroke-width:5"/>
+<circle cx="249.99999999999983" cy="148.56406460551042" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="249.99999999999983" y1="148.56406460551042" x2="309.99999999999983" y2="148.56406460551042" style="stroke:black; stroke-width:5"/>
+<circle cx="309.99999999999983" cy="148.56406460551042" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="309.99999999999983" y1="148.56406460551042" x2="289.99999999999983" y2="183.20508075688795" style="stroke:black; stroke-width:5"/>
+<circle cx="289.99999999999983" cy="183.20508075688795" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+<line x1="289.99999999999983" y1="183.20508075688795" x2="279.99999999999983" y2="165.8845726811992" style="stroke:black; stroke-width:5"/>
+<circle cx="279.99999999999983" cy="165.8845726811992" r="3" style="stroke:black; stroke-width:5" fill="black"/>
+
+</svg>