The Pythagorean Triple Wrap is inspired by the formation of 2-dimensional and 3-dimensional shapes used in Mathematics and Biology.

## Did You Know?

Understanding how polygons and other shapes work can be helpful in a variety of regular-life situations. A few examples are estimating the paint needed for a room, or calculating the square footage of your back yard or even how much mulch to order for your garden.

## How Does it Work?

The design follows a point-to-point construction (mitered square), using diagonal knitting, texture, cables, lace and short row knitting techniques. It is knitted flat in two pieces and grafted together to create a rectangular shape using four shades to add interest. The occasional use of textures, lace and cables with garter make it modern with traditional essences. It’s a good stash-busting project for both experienced and beginner knitters who love to learn new techniques and play with colors.

If you’re curious about the STEM connection, please read on. I get deep in A LOT of mathematical and biological theories. So, if you just want to make a pretty wrap, scroll down to those instructions below!

I originally named this design the polygon wrap. A *polygon *is any 2-dimensional shape formed with straight lines used to create different geometric shapes like triangles, squares, rectangles etc. A polygon is named after how many straight lines it takes to form the shape. For example, a triangle has three sides and an octagon has eight sides. Any shape that can be drawn by connecting straight lines is called a polygon.

This design shows how different geometric shapes join together to form another geometric shape. The basic mathematical concept behind this idea is “Inscribed Square in Right Triangle”. A right triangle is a triangle that has one 90 degree angle. One of the most popular right triangles is a “3-4-5 Right-triangle,” where the sides are the ratio of 3:4:5. This is also a Pythagorean Triple, as the triangle sides are in a ratio of whole numbers. According to the Pythagorean theory, for a right triangle, the sum of the squares of the two sides that join a right angle (the *legs, *a and b) equals the square of the third side (the *hypotenuse, *c).

a^{2} + b^{2} = c^{2}

With a right triangle, we can also solve many Trigonometric problems. Trigonometry helps us to find angles and distances. In a right triangle, we already know one angle, and trigonometry helps us to find the missing angles (θ) and length of sides. If we consider a square inscribed inside a right triangle, we can picture as a combination of a square and two triangles and calculate the dimensions of each shape using the Trigonometric formulas.

3^{2} + 4^{2} = 5^{2}

So how are all these theories incorporated into this design? The wrap starts from corner *a *and gradually increases to make a mitered square *abcd*. Then a triangle *bcf *is knitted on leg *bc *and triangle *cde *on the other leg *cd, *using wrap & turn short row shaping. After completing both the triangles, we find our right triangle *aef, *with hypotenuse *ef. *The wrap continues on this line, using diagonal knitting to create parallelogram *efgh. *The wrap is completed by making another right triangle *aef*, and grafting *ef *and *gh *ends together to create a rectangular shape.

So, the structure of this wrap is designed by combining different polygons (squares, triangles, a parallelogram and a rectangle) and using the basic mathematical (geometric and trigonometric) theories.

The body of the pattern also incorporates different knitting stitches and techniques inspired by mathematical and biological theories as well. A stitch pattern is used to represent the DNA Helix structure. A Helix is a curve in 3-dimensional space. Helixes are important in biology as the DNA molecule is formed as two intertwined helixes. The stitch pattern resembles the DNA double helix structure with cables (resembling a twisted knot or intertwined helix) that also relates to Knot theory. In mathematics, a *knot *is a closed curve that can be visualized as a closed loop of string. If the string has a closed knot then it would be impossible to unknot without slicing through the knot. Likewise, in biology, DNA is a complicated knot that must be unknotted by enzymes for replication or transcription. Using Knot theory, scientists can estimate how hard DNA is to unknot. It helps them to estimate the properties of enzymes that unknot DNA. Another stitch pattern used in this design is very similar to the curve lines in mathematics.

## Yarn

Knit Picks Hawthorne Tonal (80% fine superwash highland wool, 20% polyamide; 357 yds [327m]/100g): 2 skeins each of: Corvallis (A); Sweet Home (B); Astoria (C); Eugene (D).

The sample used the following approx. amounts: A: 492 yds (135g); B: 393 yds (110g); C: 535 yds (150g); D: 446 yds (125g).

## Needle

Size US 5 (3.75mm) needles, straight or 24″ (60cm) circulars

## Gauge

Garter: 23 rows/3.25 inches = 7.07 rows/in

Lace 1 (wave): 18 rows/2.625 inches = 6.8 rows/in

Lace 2 (sheaf): 32 rows/4.375 inches = 7.3 rows/in

## Size

100.5” [255 cm] long and 26” [66 cm] wide

## Notions

Tapestry needle, 1 stitch marker, stitch holder (spare needle or scrap yarn), cable needle.

## Abbreviations

See our Standard Abbreviations.

## Tutorials

## Pattern Notes

-To avoid cutting yarns, in row 193, carry contrast yarns throughout the row from behind to m1R and leave there. Continue the rest of the row with the working yarn. When working Part F, carry the contrast yarns from prev row at the beginning of each row.

-In row 234, cut the contrast yarns leaving 6” tail and continue the row with working yarn to the end. Weave yarn tails in and secure properly.

-Choosing Colors: You need 4 different colors of yarn for this wrap, though you can use more or fewer. Any color combination will give you a beautiful piece of art; just remember, this is a big wrap so choose colors that compliment your wardrobe and make you feel happy and confident. It’s a great project for digging into your stash and finding something exciting. The Color Distribution table below shows which rows are worked in which colors.

## Illustration of the Wrap

## Charts

Download the charts in PDF form (they’re large so they’re not good for a web page).

## Wrap Instructions

With A, CO 3 sts.

Work Sections 1-3 sequentially. Each consists of multiple parts with different st patterns. There are charts for all parts except A, C & H.

Sm as encountered on all rows.

## Section 1: Mitered Square

Parts A-D create a square.

### Part A

**Setup Row (WS)**: With A, k3; 3 sts.

**Row 1 (RS)**: K2, pm, m1R, k to end; 4 sts.

**Row 2**: K to m, m1R, K to end; 5 sts.

**Rows 3-14: **Rep row 2; 17 sts.

**Rows 15-24:**With B, rep Row 2; 27 sts.

**Rows 25-34:**With A, rep Row 2; 37 sts.

**Rows 35-38:**With B, rep Row 2; 41 sts.

**Rows 39-48:**With A, rep Row 2; 51 sts.

### Part B

**Row 49 (RS):**With C, k to m, m1R, k to end; 52 sts.

**Row 50 (WS):**K2, (p5, k3) 3 times, m1R, (k3, p5) 3 times, k2; 53 sts.

**Row 51:**K2, (sl5p wyf, k3) twice, sl5p wyf, k4, m1R, (k3, sl5p wyf) 3 times, k2; 54 sts.

**Row 52:**K2, (p5, k3) 3 times, p1, m1R, p1, (k3, p5) 3 times, k2; 55 sts.

**Row 53:**K4, (k1 uls, k7) 3 times, m1R, k6, (k1 uls, k7) twice, k1 uls, k4; 56 sts.

**Row 54:**K2, (p5, k3) 3 times, p2, m1R, p2, (k3, p5) 3 times, k2; 57 sts.

**Row 55:**K2, (sl5p wyf, k3) twice, sl5p wyf, k6, m1R, k5, (sl5p wyf, k3) twice, sl5p wyf, k2; 58 sts.

**Row 56:**K2, (p5, k3) 3 times, p3, m1R, p3, (k3, p5) 3 times, k2; 59 sts.

**Row 57:**K4, (k1 uls, k7) twice, k1 uls, k9, m1R, k8, (k1 uls, k7) twice, k1 uls, k4; 60 sts.

**Row 58:**K2, (p5, k3) 3 times, p4, m1R, p4, (k3, p5) 3 times, k2; 61 sts.

**Row 59:**K2, (sl5p wyf, k3) twice, sl5p wyf, k8, m1R, k7, (sl5p wyf, k3) twice, sl5p wyf, k2; 62 sts.

**Row 60:**K2, (p5, k3) 3 times, p5, m1R, p5, (k3, p5) 3 times, k2; 63 sts.

**Row 61:**K4, (k1 uls, k7) twice, k1 uls, k11, m1R, k10, (k1 uls, k7) twice, k1 uls, k4; 64 sts.

**Row 62:**K2, (p5, k3) 3 times, p6, m1R, p6, (k3, p5) 3 times, k2; 65 sts.

**Row 63:**K2, (sl5p wyf, k3) twice, sl5p wyf, k10, m1R, k9, (sl5p wyf, k3) twice, sl5p wyf, k2; 66 sts.

**Row 64:**K2, (p5, k3) 3 times, p7, m1R, p7, (k3, p5) 3 times, k2; 67 sts.

**Row 65:**K4, (k1 uls, k7) twice, k1 uls, k13, m1R, k12, (k1 uls, k7) twice, k1 uls, k4; 68 sts.

**Row 66:**K2, (p5, k3) 3 times, p8, m1R, p8, (k3, p5) 3 times, k2; 69 sts.

**Row 67:**K2, (sl5p wyf, k3) twice, sl5p wyf, k12, m1R, k11, (sl5p wyf, k3) twice, sl5p wyf, k2; 70 sts.

**Row 68:**K2, (p5, k3) 3 times, p9, m1R, p9, (k3, p5) 3 times, k2; 71 sts.

**Row 69:**K4, (k1 uls, k7) twice, k1 uls, k15, m1R, k14, (k1 uls, k7) twice, k1 uls, k4; 72 sts.

**Row 70:**K2, (p5, k3) 3 times, p10, m1R, p10, (k3, p5) 3 times, k2; 73 sts.

### Part C

**Rows 71-80:**With A, rep Row 2; 83 sts.

**Rows 81-84:**With B, rep Row 2; 87 sts.

**Rows 85-94:**With A, rep Row 2; 97 sts.

**Rows 95-104: **With B, rep Row 2; 107 sts.

**Rows 105-110: **With A, rep Row 2; 113 sts. **Rows 111-112:**With C, rep Row 2; 115 sts.

**Rows 113-114:**With A, rep Row 2; 117 sts.

**Rows 115-118:**Rep Rows 111-114; 121 sts.

### Part D

**Row 119 (RS):**With C, [K1, k2tog 2x, (yo, k1) 3 times, yo, ssk 2x] 5 times, k1, m1R, [k2tog 2x, (yo, k1) 3 times, yo, ssk 2x, k1] 5 times; 122 sts.

**Row 120 and all WS rows to 126:**K3, p to m, m1R, p to last 3 sts, k3; 1 st inc.

**Row 121:**[K1, k2tog 2x, (yo, k1) 3 times, yo, ssk 2x] 5 times, k2, m1R, k1, [k2tog 2x, (yo, k1) 3 times, yo, ssk 2x, k1] 5 times; 124 sts.

**Row 123:**(K1, k2tog, k3, yo, k1, yo, k3, ssk) 5 times, k3, m1R, k2, (k2tog, k3, yo, k1, yo, k3, ssk, k1) 5 times; 126 sts.

**Row 125:**K to m, m1R, k to end; 128 sts.

**Rows 127-128:**With A, rep Row 2; 131 sts.

** **

### Section 2: Triangles on two legs of the Square

Parts E & F, with Section 1, create a right triangle.

### Part E

**Row 129 (RS):**With C, k64, w&t.

**Row 130, 132, 134 (WS):**k to end.

**Row 131:**With A, k62, w&t.

**Row 133:**With C, k to 2 sts bef prev wrapped st, w&t.

**Rows 135-140:**With A, rep Rows 133-134.

**Rows 141-150:**With B, rep Rows 133-134.

**Rows 151-160: **With A, rep Rows 133-134. **Rows 161-166: **With B, rep Rows 133-134. **Rows 167-174:**With A, rep Rows 133-134.

**Rows 175-176:**With C, rep Rows 133-134.

**Rows 177-178:**With A, rep Rows 133-134.

**Rows 179-180:**With C, rep Rows 133-134.

**Rows 181-182:**With A, rep Rows 133-134.

**Rows 183-188:**With B, rep Rows 133-134.

**Rows 189-192:**With A, rep Rows 133-134.

**Row 193:**With C, K66, m1R, k65; 132 sts.

### Part F

**Row 194 (WS):**k60, w&t.

**Row 195 (RS):**With A, k to end.

**Row 196:**K to 3 sts bef prev wrapped st, w&t.

**Rows 197-198:**With C, rep Rows 195-196.

**Rows 199-204:**With A, rep Rows 195-196.

**Rows 205-214:**With B, rep Rows 195-196.

**Rows 215-224:**With A, rep Rows 195-196.

**Rows 225-232:**With B, rep Rows 195-196.

**Row 233:**With C, k to end.

**Row 234:**K66, m1R, k66; 133 sts.

### Section 3: Parallelogram

Parts G-K create a parallelogram on the hypotenuse diagonal line of the right triangle.

### Part G

**Row 235 (RS):**With D, kfb, k to last 2 sts, k2togtbl; 133 sts.

**Row 236 (WS):**K3, p to last 3 sts, k3.

**Row 237:**Kfb, k1, yo, ssk, p2, [(k2, yo, ssk, p2) twice] 10 times, k2, yo, ssk, p1, k2togtbl.

**Row 238:**K3, p1, yo, p2tog, [(k2, p2, yo, p2tog) twice] 10 times, k2, p2, k3.

**Row 239:**Kfb, k2, yo, ssk, p2, [(k2, yo, ssk, p2) twice] 10 times, k4, k2togtbl.

**Row 240:**K3, yo, p2tog, [(k2, p2, yo, p2tog) twice] 10 times, k2, p2, yo, k2tog, k2.

**Row 241:**Kfb, p1, k2, yo, ssk, p2, (2/2 LC, p2, k2, yo, ssk, p2) 10 times, k3, k2togtbl.

**Row 242:**K3, p1, [(k2, p2, yo, p2tog) twice] 10 times, k2, p2, yo, p2tog, k3.

**Row 243:**Kfb, p2, k2, yo, ssk, p2, [(k2, yo, ssk, p2) twice] 10 times, k2, k2togtbl.

**Row 244:**K3, [(k2, p2, yo, p2tog) twice] 10 times, k2, p2, yo, p2tog, k4.

**Row 245:**Kfb, k1, p2, k2, yo, ssk, p2, [(k2, yo, ssk, p2) twice] 10 times, k1, k2togtbl.

**Row 246:**K4, p2, yo, p2tog, [(k2, p2, yo, p2tog) twice] 10 times, k5.

**Row 247:**Kfb, k2, p2, (2/2 LC, p2, k2, yo ssk, p2) 10 times, 2/2 LC, p2, k2togtbl.

**Row 248:**k3, p2, yo, p2tog, [(k2, p2, yo, p2tog) twice] 10 times, k2, p1, k3.

**Rows 249-252:**Rep Rows 237-240.

**Row 253:**Kfb, p1, 2/2 LC, p2, (k2, yo, ssk, p2, 2/2 LC, p2) 10 times, k3, k2togtbl.

**Rows 254-258:**Rep Rows 242-246.

**Row 259:**Kfb, k2, p2, [(k2, yo, ssk, p2, 2/2 LC, p2) twice] 10 times, k2, yo, ssk, p2, k2togtbl.

**Row 260:**K3, p2, yo, p2tog, [(k2, p2, yo, p2tog) twice] 10 times, k2, p1, k3.

**Rows 261-264:**Rep Rows 237-240.

**Rows 265-266:**Rep Rows 235-236.

### Part H

**Row 267 (RS):**With B, kfb, k to last 2 sts, k2togtbl.

**Row 268 (WS):**K3, p to last 3 sts, k3.

**Row 269:**Kfb, k to last 2 sts, k2togtbl.

**Row 270:**k to end.

**Rows 271 – 288: **Rep Rows 269-270.

### Part I

**Row 289 & Row 293 (RS):**With A, kfb, k1, p1, (k17, p1) 7 times, k2, k2togtbl; 133 sts.

**Row 290 & Row 292 (WS):**K to end.

**Row 291:**Kfb, k1, p1, [(yo, k2tog) 63 times, k2, k2togtbl.

**Row 294:**K3, (k1, p17) 7 times, k4.

**Row 295:**Kfb, k2, p1, (k2tog, k6, yo, k1, yo, k6, ssk, p1) 7 times, k1, k2togtbl.

**Row 296:**K3, (ssp, p13, p2tog, k1) 7 times, p1, k3; 119 sts.

**Row 297:**Kfb, k3, p1, [k2tog, k4, (yo, k1) 3 times, yo, k4, ssk, p1) 7 times, k2togtbl; 133 sts.

### Part J

**Row 298 (WS):**K3, p14, ssp, k1, (ssp, p13, p2tog, k1) 6 times, ssp, k3; 119 sts.

**Row 299 (RS):**Kfb, k1, ssk, [p1, k2tog, k2, (yo, k1) 7 times, yo, k2, ssk] 6 times, p1, k2tog, k2, (yo, k1) 7 times, yo, k3, ssk, k2togtbl; 160 sts.

**Row 300:**K3, p18, p2tog, k1, (ssp, p17, p2tog, k1) 6 times, k4; 147 sts.

**Row 301:**Kfb, k1, ssk, (p1, k2tog, k15, ssk) 6 times, p1, k2tog, k18, k2togtbl; 133 sts.

**Row 302:**K3, (p17, k1) 7 times, k4.

**Row 303:**Kfb, k3, (p1, k17) 7 times, k1, k2togtbl.

**Row 304 and all WS Rows to 308:**K to end.

**Row 305:**Kfb, k2, p1, (yo, k2tog) 63 times, k1, k2togtbl.

**Row 307:**With C, kfb, k to last 2 sts, k2togtbl; 133 sts.

### Part K

**Row 309:**Kfb, k1, (k1, p1) to last 5 sts, k3, k2togtbl; 133 sts.

**Row 310:**K3, (p1, k1) to last 2 sts, k2.

**Row 311:**Kfb, k2, (p1, k3) to last 2 sts, k2togtbl.

**Row 312:**K3, p1, (k1, p3) to last 5 sts, k1, p1, k3.

Rep Rows [309-312] 5 more times.

** **

**Row 313:**Kfb, k to last 2 sts, k2togtbl.

**Row 314:**K to end; 133 sts.

Rep Section [3] 3 more times or to desired length, then rep Part G once more.

Transfer all sts onto a spare needle or scrap yarn for later use. Cut the working yarn leaving 6” tail and secure by weaving it in.

### Section 4: End Triangle

Work Sections 1 & 2 to create another right triangle. Keep sts on the needle and cut the working yarn leaving tail for grafting about four times the width of the piece.

## Finishing

Graft pieces together using Kitchener stitch and tapestry needle to create a stretchy and almost invisible join (see schematic). Weave in ends.

Block the finished wrap following the blocking instructions for yarn.

## About the Designer: Naurid Kashpia Senjuti

Naurid Kashpia Senjuti is a knitwear designer from Dhaka, Bangladesh. She started her career as a corporate lady but soon after her child’s birth she became a stay-at-home mom when she rediscovered her passion for crafts and hobbies. Her three-year-old super-active daughter is her inspiration for knitting, crafting and crocheting.

Ravelry designer name: Naurid Kashpia Senjuti

Website: Knit Craft Crochet

Betty

(k1 uls, k7

I don’t understand the uls abbreviation.

I’m excited to make this wrap

Thanks in advance for your help

knotions

It’s been a while so it took me a bit. It was in the chart but not the written. ULS stands for under loose stitch.

Sande Francis

I love this! I was a math major in high school, switched to accounting in college with math minor. My electives were always fun math classes. I’m already sifting thru my stash in my mind, planning to make this in worsted wt for a great warm sofa throw. Thanks so much for making it free!

knotions

Yay! We’re so glad that you’re already planning it 🙂