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We're all pretty familiar by now with the uses of the Moebius strip in knitting, and they delight my mathematical heart. But did you know that Moebius strips are just a special case, and that this category of non-orientable surfaces, ie 'objects with a twist, so with a single surface', actually exists in any dimension? The one most accessible to our limited imagination is the 4-D version, which is called a Klein bottle. It sort of looks like a bottle at first glance, except that it only has one surface, and the liquid that's in it is also outside.

It doesn't seem entirely practical for knitting. There would be room for only one sleeve in a sweater using this concept, the 2-arm version would be another topological object altogether. There'd be no place to put your hand out, irrelevant you might argue since your whole body would technically be outside. For a bag version, things you would put in would also be outside, and start by falling in between the walls, which would be rather inconvenient.

But by folding a Klein bottle right you can make a useable hat. I'm providing here a very easy version - the stitch is elementary 1x1 rib, the construction no more complicated than for a normal hat, only the assembly takes off from the ordinary. Alas, it's not perfect, it's only a 3-D representation, the surfaces should completely meld together in a moveable join, but that's always the problem with math: mere reality interferes with beautiful theory.

Still, when made in a light fuzzy yarn it also happens to looks very much like the great hat Marc Jacob is showing for next Fall, the big ball, and that should be very good for our Northern friends as it's also very warm.

Moebius strips were first introduced to knitters by Elizabeth Zimmermann in 1981, and Cat Bordhi, a Canadian mathematician, has recently published a couple of 'Magical Knitting' books that really ran away with the concept. I highly recommend them all.

model: Rosemarie Lewis photos: Marie-Christine Mahé

Adult medium
Circumference: 22 inches
Crystal Palace Merino Stripes [85% Merino/15% polyester; 115yd/105m per 50 gram ball]; Color: # 030 watermelon; 2 balls

1 set of four US #9/5.5 mm double-point needles
1 16- or 22-inch US #9/5.5 mm circular needle [optional]
3 stitch markers
2 safety pins
waste yarn
Tapestry needle

11 sts and 15 rows = 4 inches in 1x1 Rib



[Knitty's list of standard abbreviations and techniques can be found here]

If you're feeling adventurous, this hat can be worked in any stitch pattern as long as it's fully reversible (ie. it looks exactly the same on both sides). See Esther Bozak's previous work for good ideas of reversible stitches. A stitch pattern with some stretch is a good idea, since there is no separate band to hold it on your head.

The yarn chosen for this project is a self-striping yarn with a long repeat. Look at your ball of yarn carefully - which color do you prefer, or you would like next to your face? Start knitting with the other end of the ball. (If you prefer the color on the outside of the ball, start working with the end from the inside of the ball, and vice-versa.) Alternately, pick the lightest or brightest color to start with, so that the colors will shade from light to dark as they move from the top of the hat to the bottom.

For this project, you may wish to use a provisional cast on, so that the CO and BO edges may be grafted together at the end. However, unless you're an ace at grafting, this isn't strictly necessary. The hat will look just as cool if it is cast on normally and the seam will be hidden on the inside of the hat; the choice is yours.

1x1 Rib (Worked in the round over an even number of sts):
Round 1: [K1, p1] to end.
Repeat this round for 1x1 Rib.


Initial Tube
Using double-point needles, CO 12 sts. Distribute sts evenly between 3 needles (4 sts on each needle) and join to begin working in the round, being careful not to twist. Place safety pin in work to mark beginning of round. (Move the pin up every few rounds).

Work in 1x1 Rib until work measures 8 inches (or longer, if desired; this tube will form the loop which curves out from the top of the hat).

Shape Elongated Outer Cap:
Round 1: [K1, p1, m1, place marker, k1, p1, m1] 3 times. 18 sts.

6 sections have been formed, 2 on each needle. Each marker indicates the end of one section; the end of each needle indicates the end of another section.

Round 2: Work in rib as set, working new sts in pattern.

Round 3: [Work in rib as set to end of section, m1] 6 times. 6 sts increased.

Repeat Rounds 2 and 3 six times more. 60 sts.

Remove stitch markers. Retain safety pin which indicates end of round. Place a second safety pin in the last round worked. Do not move this pin up as you work; it is there for measuring purposes only. This round will be referred to as "marked round".

Note: If desired, switch to circular needles at this point, and use a stitch marker to mark end of round, instead of safety pin.

Work in 1x1 Rib as set until work measures 1 inch from marked round.

Side Opening:
A small vertical opening is formed in the side of the hat by working several rows back and forth.

Next Row: Turn work and work 1 row in pattern as set with WS of work facing you. You may wish to turn work inside out to do this.
Turn work again so that RS is facing and work 1 row.
Repeat these 2 rows once more.

Resume working in the round with RS facing.

Hat Body:
Work in 1x1 Rib as set until work measures 12 inches from marked round.

Note: If you have been using circular needles for the hat body, resume working on double-point needles at this point. Place 20 sts on each needle, and be sure that round begins with a k st.

Shape Rounded Inner Cap:
Work 1 round in rib as set, placing a marker after first 10 sts on each needle. 6 sections have been formed, as before (during outer cap shaping).

Next Round: [Ssk, work in rib as set to end of section] 6 times. 6 sts decreased.
Repeat this round 7 times more. 12 sts remain.

Work 3 rounds in rib as set, forming a very short tube.

Place remaining sts on waste yarn. Break yarn, leaving a 12-inch tail.


Push short tube inside out and to inside of work, folding work inside itself until short tube emerges from side opening.

Use yarn tail to sew held sts to CO end of tube. Be sure each st is secured. (If you chose to use a provisional CO method, graft beginning and end of work together.) Weave in ends.

Voilà! Arrange cap and handle to suit yourself, and be prepared to explain the mathematical concept to curious onlookers.


Marie-Christine's grandmother tried to teach her to knit at an early age "to shut her up". This didn't work very well on either count, but Marie-Christine took it up again later with enthusiasm (knitting, that is). After decades in San Francisco, she has recently moved to Southern France.

She has contributed to Knitty since her Vegan Fox was published in the inaugural issue. Her early mathematical career has led to a lifelong interest in strange objects, many of which are discussed at