Fractals are geometric patterns that show self-similarity, they can be very complex and really beautiful.
You can also bead fractals. This is a pair of earrings that uses a very simple fractal pattern to create a delicate fringe or tassel that resembles some of the fractals you can see in nature like trees or the veins in your body.
The shape I used was 3 lines. Each time you add another set of 3 lines to the ends of the lines, that's called an iteration. This animation shows 4 iterations of the shape I used.

Thread isn't rigid, so when you create the same shape in beads, you get the earrings above.
You will need:
4 colors of size 15 Japanese seed beads (a, b, c, d)
a stop bead- I used a random seed bead that wasn't one of the colors I was working with.
earring wires
thread- I used Silamide size A, but if you want something a little firmer, try a fine monofilament that's a bit wiry.
scissors,
needles, I used a size 12 needle and it worked fine with the thread passes
You need to decide the order you want the beads in, you can absolutely use just one color for them, but that will mean a lot more counting. Using 4 colors means less counting so I use 3 shades of blue and a frosted s/l clear.
String the delica, 24 of color A, 12 color B, 6 color C and 3 color D. As you can see, each color decreases by half.
Go back up through the second bead from the end, and then the one after that, add 3 more color D beads, go back through the second bead from the end and the one after that, add one more 3 bead fringe the same way.

Now you've finished the first set of beads on the last iteration of the fractal. Go back up the 6 color C beads, and add 6 color C and 3 color D and create that leg. If you click on the image below, you'll see a full size schematic of the bead layout, hit your browser's back button to come back here.

So you understand the basic idea right? 3 lines, each has 3 lines coming off of it, repeated to 4 iterations just like the animation above. So after you finish the first line, pass your needle and thread back through the Delica in the middle.Slide off the stop bead and tie the two thread ends in surgeon's knot. then do the second "leg".

Finish that line off the same way you did the first one, and pass your needle and thread back up and through the Delica from the opposite side of the one you came out of to start that leg.

When you thread back through to the beginning Delica, you can put your accent beads over that and attach your earring wire using whichever method you prefer. I used 2 bead caps and a seed bead loop on mine. Weave in the thread ends.
Making fractals different sizes using this method is easy. Just remember that each iteration is half the size of the one before it, so it's easier to work out the math for it by starting with the smallest iteration. Like if you want 4 beads at the end and 3 iterations, it would be 4-8-16 or 16 beads for the first color, 8 beads for the second color, and 4 beads for the last. You can also increase the number of starting spokes. The design will get increasingly complex as you add spokes. I think it would be a wonderful tassel necklace using 4 lines to begin with and 3 iterations in size 11 seed beads.

If you want to know more about fractals in nature, I can't recommend this book enough.

Tessellated Skull Pattern
Knotwork Patterns E-book
Graphing Snowflake Patterns

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