Pentagons or things start to get tricky

The basic flavours remain the same: geometric or swirl but a swirl can become a geometric and even the geometric might not work as described. If you don't like reading the word "experiment" and if "might not be needed" strikes terror in your heart you aren't pentagon material.

The pentagon polygon is the point where the number of needles available can become an issue. If you do not have six needles to work with, arrange the stitches on 3 needles in the following configuration 2, 4 (2 marker 2) & 4 (2 marker 2) and knit with a 4th needle. The markers on the two needles that hold two sides of the pentagon will delimit your "virtual" needles.

The Geometric Pentgon

To knit a Geometric Pentagon -- cast on 10 stitches, divide evenly onto five needles, join, do not twist, knit with a sixth needle.
Round 1: Knit

Round 2: Increase into every 2nd stitch (15 stitches 3 on each needle)

Round 3: Knit

Round 4: Increase at the beginning and end of all 5 needles (25 stitches 5 on each needle)

Rounds 5 & 6: Knit

Round 7: Repeat Round 4

Round 8: Knit

Repeat from Round 4

In the example, on each increase round, I used a yarn over, knit to last stitch, yarn over knit 1 on each needle.
That's right, as given the instruction is 2 increase rounds in every 5 rounds with the caveat that YMMV and you may or may not need the extra no increase round.
In my experience, omitting the two extra no increase rounds you get a 3-dimensional shaped pentagon. Omitting the third rest round seems to produce a flat shape at least at my gauge so 1 increase rounds in every 3 rounds may work.

The Swirl Pentagon

To knit a Swirl Pentagon -- cast on 10 stitches, divide evenly onto five needles, join, do not twist, knit with a sixth needle.
Round 1: Knit

Round 2: Increase at the beginning of all 5 needles (20 stitches 4 on each needle)

Repeat Rounds 2

In an interesting notation here, Mary Thomas suggests that if the increase method used is what she calls an O.2 (or a double yarn over) failure to drop the 2nd over on the next round will result in a Geometric Pentagon rather than a Swirl. This suggests that perhaps a Geometric Pentagon could be produced with in a 1 increase round and one non-increase round rather than the 2 in 5 or 2 in 4 noted above but only if this specific increase method is used.

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The Geometric Pentgon
	To knit a Geometric Pentagon -- cast on 10 stitches, divide evenly onto five needles, join, do not twist, knit with a sixth needle. Round 1: Knit Round 2: Increase into every 2nd stitch (15 stitches 3 on each needle) Round 3: Knit Round 4: Increase at the beginning and end of all 5 needles (25 stitches 5 on each needle) Rounds 5 & 6: Knit Round 7: Repeat Round 4 Round 8: Knit Repeat from Round 4 In the example, on each increase round, I used a yarn over, knit to last stitch, yarn over knit 1 on each needle. That's right, as given the instruction is 2 increase rounds in every 5 rounds with the caveat that YMMV and you may or may not need the extra no increase round. In my experience, omitting the two extra no increase rounds you get a 3-dimensional shaped pentagon. Omitting the third rest round seems to produce a flat shape at least at my gauge so 1 increase rounds in every 3 rounds may work.
The Swirl Pentagon
	To knit a Swirl Pentagon -- cast on 10 stitches, divide evenly onto five needles, join, do not twist, knit with a sixth needle. Round 1: Knit Round 2: Increase at the beginning of all 5 needles (20 stitches 4 on each needle) Repeat Rounds 2 In an interesting notation here, Mary Thomas suggests that if the increase method used is what she calls an O.2 (or a double yarn over) failure to drop the 2nd over on the next round will result in a Geometric Pentagon rather than a Swirl. This suggests that perhaps a Geometric Pentagon could be produced with in a 1 increase round and one non-increase round rather than the 2 in 5 or 2 in 4 noted above but only if this specific increase method is used.