This is the most easily created of the wearable Sierpinski variations. I've made several of these, and the pattern comes out quite strikingly. Gauge and other factors are not guaranteed, and you may need to modify this pattern to make something that works with your crochet style.
This used to be called the "Studio 90" hat: I named my hypothetical design studio after the Wolfram-enumeration number of the automaton which produces Sierpinski triangles. Unfortunately, there's already a boutique by that name, so in the interest of not getting kicked in the head by trademark lawyers, I've elected to change the number to that of a somewhat different automaton.
You are free to use this pattern for your own personal use, including for your own consumption and for gifts. This pattern is not licensed for redistribution (except, unaltered, for educational purposes) or commerical purposes; if you wish to use it in either of those capacities, please contact me for terms of special licensing. Use of the underlying motif for purposes other than hats is cheerfully encouraged, and such products or patterns are free for commercial use or redistribution, as long as credit is given for the original motif in any derivative patterns.
All stitches in this hat are worked either in intersticial spaces or around posts. Unless otherwise stated, each stitch is worked in the next available space (in the case of hdc) or around the next available post (in the case of all the fp stitches). The following are all the stitches you should need to complete the hat:
Half double crochet (hdc): Yarn over, insert hook into space between two stitches, draw up a loop, yarn over, and pull through all 3 loops on the hook.
Front post double crochet (fpdc): Yarn over, insert hook behind post of next stitch, draw up a loop, and (yarn over, pull through 2 loops on the hook) twice.
Front post double cluster (fpdc2tog; uses next two posts): (Yarn over, insert hook behind post of next stitch, draw up a loop, yarn over, and pull through 2 loops on the hook) twice. Yarn over and pull through all 3 loops on the hook.
Using an I or J hook and worsted-weight or lighter yarn, ch 3 and join to form a loop.
Round 1: ch 1, 6 hdc in loop, join to first hdc.
Round 2: ch 1, skip current space, 2 hdc in each space
around, join to first hdc: 12 hdc.
Round 3: ch 1, skip current space, (2 hdc in next space,
hdc in next space) around, join to first hdc: 18 hdc.
Round 4: ch 1, skip current space, (hdc in next space, 2
hdc in next space, hdc in next space) around, join to first hdc: 24
hdc.
Round 5: ch 1, skip current space, (2 hdc in next space,
hdc in next 3 spaces) around, join to first hdc: 30 hdc.
Round 6: ch 1, skip current space, (hdc in next 2 spaces, 2
hdc in next space, hdc in next 2 spaces) around, join to first hdc: 36
hdc.
Round 7: ch 1, skip current space, (hdc in next 4 spaces, 2
hdc in next space, hdc in next 4 spaces) around, join to first hdc: 40
hdc.
Round 8: ch 1, skip current space, (2 hdc in next space,
hdc in next 9 spaces) around, join to first hdc: 44 hdc.
Round 9: ch 1, skip current space, (hdc in next 5 spaces, 2
hdc in next space, hdc in next 5 spaces) around, join to first hdc: 48
hdc.
Round 10: ch 1, skip current space, (2 hdc in next space,
hdc in next 11 spaces) around, join to first hdc: 52 hdc.
Round 11: ch 1, skip current space, (hdc in next 6 spaces, 2
hdc in next space, hdc in next 6 spaces) around, join to first hdc: 56
hdc.
Round 12: ch 1, skip current space, (2 hdc in next space,
hdc in next 13 spaces) around, join to first hdc: 60 hdc.
Round 13: ch 1, skip current space, (2 hdc in next space,
hdc in next 19 spaces) around, join to first hdc: 63 hdc.
Round 14: ch 1, skip current space, (hdc in next 10 spaces, 2
hdc in next space, hdc in next 10 spaces) around, join to first hdc: 66
hdc.
Round 15: ch 1, skip current space, (2 hdc in next space,
hdc in next 21 spaces) around, join to first hdc: 69 hdc.
Round 16: ch 1, skip current space, * hdc in next space,
(fpdc2tog, hdc) 7 times, fpdc2tog, hdc in next 3 spaces, skip next
space, 2 fpdc around next post, skip next space, hdc in next 2
spaces. Repeat twice from *, join to first hdc: 24 fpdc2tog, 39 hdc,
and 6 fpdc.
Round 17: ch 1, skip current space, * hdc in next space, (fpdc,
skip next post, fpdc, hdc in next 2 spaces) 3 times, fpdc, skip next
post, fpdc, hdc in next 3 spaces, skip next space, fpdc, hdc, fpdc,
skip next space, hdc in next 2 spaces. Repeat twice from *, join to
first hdc: 30 fpdc and 39 hdc.
Round 18: ch 1, skip current space, * hdc in next space,
(fpdc2tog, hdc in next 3 spaces) 3 times, fpdc2tog, hdc in next 3
spaces, skip next space, 2 fpdc around next post, skip next post,
2 fpdc around next post, skip next space, hdc in next 2 spaces. Repeat
twice from *, join to first hdc: 12 fpdc2tog, 45 hdc, and 12 fpdc.
Round 19: ch 1, skip current space, * hdc in next space, fpdc,
skip next space, hdc in next 2 spaces, skip next space, fpdc, hdc in
next 4 spaces, fpdc, skip next space, hdc in next 2 spaces, skip next
space, fpdc, hdc in next 3 spaces, skip next space, fpdc, hdc in next
3 spaces, fpdc, skip next space, hdc in next 2 spaces. Repeat twice
from *, join to first hdc: 18 fpdc and 51 hdc.
Round 20: ch 1, skip current space, * hdc in next space,
fpdc2tog, hdc, fpdc2tog, hdc in next 5 spaces, fpdc2tog, hdc,
fpdc2tog, hdc in next 3 spaces, skip next space, 2 fpdc around next
post, skip space, hdc in next 2 spaces, 2 fpdc around next post, hdc
in next 2 spaces. Repeat twice from *, join to first hdc: 12 fpdc2tog,
45 hdc, and 12 fpdc.
Round 21: ch 1, skip current space, * hdc in next space, fpdc,
skip next post, fpdc, hdc in next 6 spaces, fpdc, skip next post,
fpdc, hdc in next 3 spaces, skip next space, fpdc, hdc, fpdc, skip
next space, hdc, skip next space, fpdc, hdc, fpdc, skip next space,
hdc in next 2 spaces. Repeat twice from *, join to first hdc: 24 fpdc
and 45 hdc.
Round 22: ch 1, skip current space, * hdc in next space, fpdc2tog,
hdc in next 7 spaces, fpdc2tog, hdc in next 3 spaces,
skip next space, (2 fpdc around next post, skip next post) 3 times,
2 fpdc around next post, skip next space, hdc in next 2 spaces. Repeat
twice from *, join to first hdc: 6 fpdc2tog, 39 hdc, and 24 fpdc.
Round 23: ch 1, skip current space, * hdc in next space, fpdc,
skip next space, hdc in next 6 spaces, skip next space, fpdc, hdc in
next 3 spaces, skip next space, fpdc, hdc in next 7 spaces, fpdc, skip
next space, hdc in next 2 spaces. Repeat twice from *, join to first
hdc: 12 fpdc and 57 hdc.
Round 24: ch 1, skip current space, * hdc in next space,
(fpdc2tog, hdc) 3 times, fpdc2tog, hdc in next 3 spaces, skip next
space, 2 fpdc around next post, skip space, hdc in next 6 spaces, skip
space, 2 fpdc around next post, skip space, hdc in next 2
spaces. Repeat twice from *, join to first hdc: 12 fpdc2tog, 45 hdc,
and 12 fpdc.
Round 25: ch 1, skip current space, * hdc in next space, fpdc,
skip next post, fpdc, hdc in next 2 spaces, fpdc, skip next post,
fpdc, hdc in next 3 spaces, skip next space, fpdc, hdc, fpdc, skip
next space, hdc in next 5 spaces, skip next space, fpdc, hdc, fpdc,
skip next space, hdc in next 2 spaces. Repeat twice from *, join to
first hdc: 24 fpdc and 45 hdc.
Round 26: ch 1, skip current space, * hdc in next space,
fpdc2tog, hdc in next 3 spaces, fpdc2tog, hdc in next 3 spaces, skip
next space, 2 fpdc around next post, skip next post, 2 fpdc around
next post, skip next space, hdc in next 4 spaces, skip next post, 2
fpdc around next post, skip next post, 2 fpdc around next post, skip
next space, hdc in next 2 spaces. Repeat twice from *, join to first
hdc: 6 fpdc2tog, 39 hdc, and 24 fpdc.
Round 27: ch 1, skip current space, * hdc in next space, fpdc,
skip next space, hdc in next 2 spaces, skip next space, fpdc, hdc in
next 3 spaces, skip next space, fpdc, hdc in next 3 spaces, fpdc, skip
next space, hdc in next 3 spaces, skip next space, fpdc, hdc in next 3
spaces, fpdc, skip next space, hdc in next 2 spaces. Repeat twice from
*, join to first hdc: 18 fpdc and 51 hdc.
Round 28: ch 1, skip current space, * hdc in next space,
fpdc2tog, hdc, fpdc2tog, hdc in next 3 spaces, skip next space, (2
fpdc around next post, skip next space, hdc in next 2 spaces, skip
next space) 3 times, 2 fpdc around next post, skip next space, hdc in
next 2 spaces. Repeat twice from *, join to first hdc: 6 fpdc2tog, 39
hdc, and 24 fpdc.
Round 29: ch 1, skip current space, * hdc in next space, fpdc,
skip next post, fpdc, hdc in next 3 spaces, skip next space, (fpdc,
hdc, fpdc, skip next space, hdc, skip next space) 3 times, fpdc, hdc,
fpdc, skip next space, hdc in next 2 spaces. Repeat twice from *, join
to first hdc: 30 fpdc and 39 hdc.
Round 30: ch 1, skip current space, * hdc in next space,
fpdc2tog, hdc in next 3 spaces, skip next space, (2 fpdc around next
post, skip next post) 7 times, 2 fpdc around next post, skip next
space, hdc in next 2 spaces. Repeat twice from *, join to first
hdc: 3 fpdc2tog, 48 fpdc and 18 hdc.
Round 31: ch 1, skip current space, hdc in each space around,
join to first hdc: 69 hdc.
Repeat round 31 to desired length and finish off.
Rounds 1-15 are the crown of the hat. You can substitute any crown design you like, as long as its last round has 69 stitches.
Rounds 16-30 are the Sierpinski-triangle patterns. Following these can be confusing: there's a set of rules which makes a great deal of sense on the inside of my head, but are hard to translate into words. There are, however, certain general rules you can apply to see if you're doing the design right: an fpdc is always, except on round 16, built on the post of another fpdc or a fpdc2tog. Similarly, on all rounds except 16, 20, 24, and 28, an fpdc2tog is always built on the post of two fpdcs.
You can of course use this design on anything worked in the round, or even on back-and-forth work, although if worked back-and-forth half of the front-post stitches must be changed to back-post stitches. Also, the number of repetitions of the pattern, and the spacings used, can be modified for the best fit on your particular hat-sizing. The design as listed above includes 3 repetitions of the motif (with two triangles in each), with spacings of 3 between the triangles. Since the triangles themselves use 17 stitches per pair, the number of stitches used is r(17+2s), where s is the number of spacing stitches, and r is the number of motif repetitions. These design modifications can be easily engineered by replacing the end-of-triangle instruction "hdc in next 3 spaces" with "hdc in next s spaces", and repeating the motif r times instead of 3. Some useful numbers of stitches you can use are:
You're rather on your own in devising a crown with the correct number of stitches though.