Archive for the ‘Knitting’ Category

Another Coded Blessing Blanket!

Saturday, March 11th, 2017

I finished another Mathematical Knitting Project!

This is another Coded Blessing Blanket.

This time, though, it’s for my 22-year-old son.

I’d finished knitting for babies in the family. (Though now it’s time to knit again.) So I asked my son if there were anything he’d like me to knit for him. He said, “Blue Blankie could use a stunt double.”

Blue Blankie is the blanket I knitted him when I was expecting his birth and I was on bed rest. After he was born, I gave him the blanket every time I fed him. I was happy when he took Blue Blankie with him to college, but yes, sadly, Blue Blankie is falling apart.

However, Tim said he’d like a purple blanket this time. And I’d already used the same pattern to knit a blanket for my niece Alyssa with a blessing coded in the stitches. So I decided to do the same for Tim.

Here’s how it works. The stitches make a sort of plaid pattern with knits and purls. The pattern has a sequence of 12 rows that make one large pattern-row. Each pattern-row has seven smooth panels on the front side of the blanket. And each smooth panel is split into two parts. So I used those panels to code words of seven letters or less.

The code I used was base 5. So A = 01, B = 02, C = 03, D = 04, E = 10, F = 11, and so on. So I only need 5 stitch patterns, using three stitches.

I used some simple patterns. 0 is knit each stitch. This matches the background.

1 is purl each stitch, making a bumpy row.

2 is a cable made by holding 2 stitches to the back.

3 is a cable made by holding 1 stitch to the front. (Making the cable go the opposite direction from 2.)

4 is a yarnover and knit 2 together — making a hole.

Here’s a closer look at how the stitches turned out.

And an even closer look.

The first four rows are my son’s name, Timothy Ronald John Eklund. So the first row, for example, is 40 14 23 30 40 13 100. (To knit Y, I began one stitch ahead of the 7-stitch panel, using 9 stitches for 100.)

I’m not going to tell what the rest of the blanket says, except to say it’s a blessing. Can Tim read the code?

Now, this is exactly the same way I made Alyssa’s Blessing Blanket, but it turned out that hers had an error. I had almost finished Tim’s at Christmas time, but finally proofread it — and found an error, took out about 50 rows, and reknitted them. But now it’s done, and it’s error-free!

And it was a wonderful thing to knit a blanket full of love for my 22-year-old son who had just moved to the other side of the country.

May you thrive, Tim!

Review of A Hat for Mrs. Goldman, by Michelle Edwards, illustrated by G. Brian Karas

Thursday, February 2nd, 2017

A Hat for Mrs. Goldman

A Story About Knitting and Love

by Michelle Edwards
illustrated by G. Brian Karas

Schwartz & Wade Books, 2016. 36 pages.
Starred Review
2016 Sonderbooks Stand-out: #5 Picture Books
2017 Sydney Taylor Book Award Silver Medalist

Oh, here is a picture book for knitters to love!

Unlike many stories about knitting, it acknowledges that knitting is difficult and takes a long time. And this ends up being a beautiful story about showing love by knitting.

Mrs. Goldman knits hats for the whole neighborhood, including Sophia.

“Keeping keppies warm is our mitzvah,” says Mrs. Goldman, kissing the top of Sophia’s head. “This is your keppie, and a mitzvah is a good deed.

Sophia goes with Mrs. Goldman when she walks her dog Fifi, and Sophia notices that Mrs. Goldman doesn’t have a hat any more. She gave it to Mrs. Chen.

Sophia gets an idea.

Last year, Mrs. Goldman taught Sophia how to knit.
“I only like making pom-poms,” decided Sophia after a few days.
“Knitting is hard. And it takes too long.”

Now Sophia digs out the knitting bag Mrs. Goldman gave her. And the hat they started.
The stitches are straight and even. The soft wool smells like Mrs. Goldman’s chicken soup.

Sophia holds the needles and tries to remember what to do. She drops one stitch. She drops another.

Still Sophia knits on. She wants to make Mrs. Goldman the most special hat in the world.

Sophia works hard on that hat. For a long time. Finally she finishes knitting and sews it up.

I love that the hat doesn’t look very good. In fact, it looks like a monster hat.

But Sophia’s solution is wonderful, and fits with what went before. She covers the hat with red pom-poms. When she gives it to Mrs. Goldman, she says it reminds her of Mr. Goldman’s rosebushes.

And now her keppie is toasty warm. And that’s a mitzvah.

The book finishes up with instructions for knitting a simple hat and for making pom-poms.

(Hmmm. Now as I post this, I think it’s pretty much a Pussy hat. But you can cover it with pom-poms if you like. Or not.)

This is a beautiful story, as it says, about knitting and love.

Buy from

Find this review on Sonderbooks at:

Disclosure: I am an Amazon Affiliate, and will earn a small percentage if you order a book on Amazon after clicking through from my site.

Source: This review is based on a library book from Fairfax County Public Library.

Disclaimer: I am a professional librarian, but I maintain my website and blogs on my own time. The views expressed are solely my own, and in no way represent the official views of my employer or of any committee or group of which I am part.

What did you think of this book?

Normal Distribution Scarf

Friday, August 12th, 2016

Today I finished a second Normal Distribution Scarf.


The first one I made was designed to highlight outliers to show that outliers are what makes the world beautiful.

For this one, I only wanted to show the Normal Distribution. I decided to knit it the long way so this time I wouldn’t have to sew any ends in.

I took colors from light to dark, in shades of pink. Colors B and C were a little closer than I wanted them to be, but it still gave the idea.


I generated numbers from a normal distribution and made a big list. For positive values, I purled the row, and for negative values, I knitted — so those values should be about even, making random ridges.

For the color, I used the absolute value, from light to dark. Since the normal distribution is a bell curve, there should be many more values in the lighter colors.

For 0 to 0.5, I used White.
0.5 to 1.0 was Victorian Pink.
1.0 to 1.5 was Blooming Fuchsia (only a little darker than Victorian Pink).
1.5 to 2.0 was Lotus Pink — a bright, hot pink.
Above 2.0 was Fuchsia — a dark burgundy.


Naturally, I used a lot more of the lighter colors. So for my next project after my current one, I think I’m going to do another normal distribution scarf, but this time reversing the values. So the new scarf would be mainly dark colors with light highlights.

In fact, if I weren’t using pink (maybe purple or blue), it would be fun to make scarves for a couple this way. Use dark, staid, sedate colors for the man, with light highlights. Use pastel shades for the woman — with dark highlights. [Hmmm. If I knit a scarf for a boyfriend before he exists, would the boyfriend jinx not apply?]


In this version, the lighter colors were more prominent.

Here’s a view of the scarf draped over my couch, showing both sides.


The different look has to do with where the knits and purls were placed and which side has a ridge and which is smooth.

Here’s a closer look:


I like the way the color combinations turned out so pleasing.


The only real problem is that the scarf is made out of wool, and it was almost 100 degrees outside today. So for now, I’m going to have to enjoy it draped over my couch rather than wearing it. I’ll look forward to this summer!


Coded Affirmations Scarf

Saturday, February 27th, 2016

affirmations_scarfThe one knitted object in my Sonderknitting Mathematical Knitting Gallery which I haven’t explained is the Coded Affirmations Scarf.

I knitted the scarf with a small ball of leftover yarn before I knitted Alyssa’s Coded Blessing Blanket, but after it had occurred to me that you could use mathematical bases to make coded messages, as in this Base Six Code Coloring Sheet. With knitting, instead of colors, you use a different two-stitch stitch pattern for the code.

I’m pretty sure I used a similar code for the Affirmations Scarf as I did for the Blessing Blanket later. The patterns would have involved knit and purl stitches, cables to the front or back, and yarn overs with decreases. But to be honest, the scarf is much harder to read because it’s not as clear where the letters begin and end. (It was nice in the blanket that I had a built-in grid to use.)

Anyway, the idea wasn’t to be able to decipher it. The idea was that I would know what the scarf said.

What does the scarf say? My name — my full name, my nickname — and words that I believe describe me. (Along the lines of “Loved,” “Joyful” — you get the idea.)

And the effect is just a seemingly random lacy pattern.

This was my first experiment. And looking at it now, years later — well, I still haven’t been able to decipher it. (I’m hoping I wrote down the code somewhere!)

But the idea — to knit meaning into a scarf with a coded message — was a complete success.

And I still say you could do this with colors on the edge of a picture or anywhere else you want a secret meaning hidden in a pretty pattern.


My posts on Mathematical Knitting and related topics are now gathered at Sonderknitting.

Jade’s Finished Outliers Scarf!

Thursday, February 4th, 2016

Today I finished the Normal Distribution Scarf I made for my transgender daughter Jade!


This scarf shows that it is the outliers that make life beautiful.

A lot of things in life have a normal distribution — height, intelligence, and many other things. Most people are somewhere near the middle of the bell-shaped curve.

All her life, Jade has had qualities that are outliers. And I do believe that has much to do with why she is such a beautiful person. She definitely adds spice to life!

Here’s how I made the scarf:

I chose four colors of yarn. Then I generated random numbers from a normal distribution. I used the website


The numbers told me what colors to use for each row.

OutliersScarfLengthIf the number was negative, I knitted. If it was positive, I purled. (This will be about even for each.)

For numbers from -0.5 to 0.5, I used brown, Color A.

For numbers from -1.0 to -0.5 and 0.5 to 1.0, I used a brownish burgundy, Color B.

For numbers from -1.5 to -1.0 and 1.0 to 1.5, I used bright red, Color C.

For numbers less than -1.5 and bigger than 1.5, I used a rainbow yarn, Color D.

The rainbow yarn changed only gradually. It started out orange and gradually changed to yellow, then green, then pink. But this yarn for the outliers definitely is the most noticeable yarn throughout.

The only thing I didn’t like about this scarf is that there were far too many ends to sew in, and I didn’t feel like I did a great job of covering that up with a crocheted edging. If I make a normal distribution scarf again, I will probably knit it lengthwise, even though that won’t use as many numbers.

I was also thinking I’d like to use an additional color for 1.5 to 2.0. Then the outliers yarn would be more rare. I also might try using an amount of 0.75 for each section instead of 0.5, so that the sections would be 0 to 0.75, 0.75 to 1.5, and 1.5 to 2.25.

I’m going to test these two ideas on a coloring sheet before I try knitting another scarf.

You can find various more mathematical knitted objects and coloring sheets at

Zoe’s Prime Factorization Blanket!

Saturday, January 16th, 2016

Back in November, I finished my little niece Zoe’s Prime Factorization Blanket!


What is a Prime Factorization Blanket? Why, a blanket that shows the prime factorization of all the whole numbers up to 99, using a color for each prime number.

This is the same set-up as my niece Arianna’s Prime Factorization Blanket, as a matter of fact. But I used new colors for Zoe’s blanket, going with a lot of pink, because we already knew she was going to be a girl. (With Arianna, we found out she’d be a girl right when I got to the number 17, so in that blanket 17 is pink.)

The blankets don’t really need a pattern, but here are the specifications: I used Tahki Cotton Classic yarn, because it has so many shades available. Each square is a garter stitch square with 12 ridges and 12 stitches, which is easy to divide in 2, 3, 4, or 6 sections. For 5 sections, I did a plain row at the beginning and end. It’s done in entrelac, so you go across and knit the square for each number individually, then go back making the white squares, then do the next row of numbers, then a row of white. It’s much nicer than making the original sweater, because you can work on one number at a time, and don’t have to carry yarn across.

Here is Zoe’s Prime Factorization Blanket laid out flat (or sort of flat):

Zoes Blanket

Here’s how it works. Starting in the bottom left corner (because graphs always have the origin in the bottom left), there’s a missing space for zero. Then 1 is pale pink, the background color:

Zoes Blanket bottom left

2 was assigned the color pink.
3 was assigned the color red.

Zoes Blanket bottom middle

4 is our first composite number, 2 x 2. So I used two sections of pink. (If you look at the actual blanket, you can tell there are two sections, but it’s harder to tell in the picture.)
5 is prime, so it’s assigned a new color, yellow.
6 is composite, 2 x 3. So it gets a section of pink and a section of red.

Zoes Blanket bottom right

7 is prime, so it gets a new color, purple.
8 is composite, 2 x 2 x 2. Three sections of pink.
9 = 3 x 3, so it gets two sections of red.

New row, so look back at the photo of the bottom right.
10 = 2 x 5, so it gets a section of pink and a section of yellow.
11 is prime, so it gets a new color, turquoise.
12 = 2 x 2 x 3, so two sections of pink and one section of red.
13 is prime, so it gets a new color, sea foam green.

Now the picture for the middle:
14 = 2 x 7, so pink and purple.
15 = 3 x 5, so red and yellow.
16 = 2 x 2 x 2 x 2, so four sections of pink.

Now the picture of the right side:
17 is prime, so it gets a new color, baby blue.
18 = 2 x 3 x 3, one section of pink, two sections of red.
19 is prime, so it gets a new color, olive green.

The next row starts at 20. The blanket goes all the way up to 99.

Here’s the top corner, so you can see some bigger numbers:

Zoes Blanket top corner

You can see the patterns nicely in the grid of the blankets. As an example of some simple patterns, the twos and fives line up in straight lines, but so do the elevens, in a diagonal line. There are lots more patterns which you can find the more you look at the blanket.

And Zoe likes it!


I’m gathering all my Mathematical Knitting links on my Sonderknitting page. (I hope to soon add coloring pages, too!) Check out the rest!

An Outliers Scarf for Jade

Wednesday, December 2nd, 2015


I recently posted an explanation of my Probability Scarf, where I simply rolled a die to decide which of 6 colors to use for each row of the scarf.


But that represents a uniform distribution, where each color is equally likely — a little boring.

So I thought: Why not make a scarf using the normal distribution, a bell-shaped curve. I searched the web and found a site that would give me random numbers generated from a normal distribution.

I’ll use four colors:


Brown is for the center of the distribution (numbers within half a standard deviation from the mean). This is where most of the data will fall.

The next color has a bit more red in it, but it’s between red and brown. This will be for numbers between a half and one standard deviation from the mean.

The third color will be used for numbers more than one standard deviation from the mean, but less than one and a half standard deviation. It’s quite bright and red and pretty.

And finally — for the outliers — I bought a rainbow yarn. It turns out it changes colors very slowly, so you can’t necessarily tell that it’s rainbow-colored in the scarf, but it is bright and is slowly changing.

Also, about half the numbers are negative and half positive. I went with positive is for purl and negative is for knit.

And the point of the scarf? It is the outliers that make it beautiful! Yes, we need the nice middle-of-the-road, close to the mean folks — but the colorful ones are the outliers and add spice to life.

I’m planning to give the scarf to my daughter Jade, who has always been an outlier in several areas — and I fully believe that has a lot to do with why she is so wonderful.


The scarf is turning out lovely. I plan to continue until I run out of one color. (I bought two skeins of the brown yarn.) Yes, I am going to have lots of ends to sew in when I am done! I’m planning to do a crocheted edging in brown to cover up some of that.


I’m gathering all my Mathematical Knitting links on my Sonderknitting page.

Fibonacci Swatchy

Friday, November 27th, 2015

My sister-in-law is expecting a baby next June. Her toddler already has a Prime Factorization Blanket, and I just finished making a second one for a niece in another family. It’s time for something new!

Inspired by my Fibonacci Clock (not my idea, but a clock purchased via Kickstarter) and my Fibonacci Spiral Earrings, I’m thinking about making a Fibonacci Spiral Blanket.

Fibonacci Clock

The Fibonacci Sequence is simple. You start with 1, then each new number is the sum of the two numbers before it:

1 + 0 = 1
1 + 1 = 2
1 + 2 = 3
2 + 3 = 5
3 + 5 = 8
5 + 8 = 13
8 + 13 = 21
and so on. . .

I made a swatch to see if it would work, and I think it’s going to. Here’s the Fibonacci Swatchy:


It starts with the little white square, which represents 1. I planned to make the blanket 12 stitches by 12 garter ridges. I made the swatch 6 by 6, and think I may go with that for the blanket after all. The important thing is for it to be divisible by 3. It’s going to get big fast.

Okay, after the initial square, I picked up stitches along one edge of the square. I added a new color for this square, but it’s the same size as the first, still representing 1. Since 1 = 1 + 0, I used the first color (white), but added a new color representing the new entry in the sequence.

For the next square, representing 2, I picked up 12 stitches along both the previous squares. I use three colors — representing the two numbers whose sum in the new entry. This pattern will continue. Each new Fibonacci number will get a new color of its own — but I’ll alternate that with the two colors representing the two numbers I summed to get this number.

And in garter stitch it turned out very cool if you alternate rows of three colors — It turns out that you will have the yarn waiting for you when you’re ready to pick up that color again on the correct side. And the garter ridges work out to look like solid stripes. There are two colors in between the ridges, but because of the way the texture works, you see the matching color ridges together.

So in the swatch, the entry representing 2 was a 12 by 12 square alternating white, pink, and burgundy.

For the next entry, representing 3, I picked up stitches along the square I just finished plus one of the 1 squares, so that made 18 stitches, and I went for 18 rows. I dropped the first color white, and now alternated pink, burgundy, and a new color, lavender.

To finish it off, I chain stitched in a golden Fibonacci spiral. For the actual blanket, I’ll be a little more careful to make each curve circular.

I think this may make a fine blanket. The squares will get big quickly, so I’m not sure how far it will go. My brother and his wife should find out the baby’s gender in January. Though I’m thinking even if the baby is a girl, I may want to use more gender-neutral colors in the middle (these starting squares) and save pink for the bigger squares that will come later. But we’ll see. I also learned a little bit by swatching about how I want to pick up the stitches. But the main lesson is that alternating three colors in garter stitch works great! And crocheting on a golden spiral works great!

This is going to be fun!

My posts on Mathematical Knitting and related topics are now gathered at Sonderknitting.

My Probability Scarf

Wednesday, October 7th, 2015


I’ve started collecting my Mathematical Knitting posts at Sonderknitting, a Mathematical Knitting Gallery.

But I’d never done a post about my Probability Scarf.

This is not my idea. I don’t remember where I saw the instructions, but they are easy and a lot of fun.

1. Choose six colors of yarn that go together well. Assign them numbers from 1 to 6.

I chose leftovers from my Prime Factorization Sweater.

2. You’ll be knitting a scarf the long way, using the ends as fringe. Start by casting on to a circular needle however long you want your scarf to be. (Try to keep it loose!)

3. For each row, roll a die to decide which color to use. Flip a coin to decide whether to knit or purl.

4. Continue in this manner until you’ve run out of one of the colors.

You now have a scarf demonstrating the Uniform Distribution.

This scarf was fun to knit. It was hard to stop knitting, because I kept wondering what the next row would look like.

It occurs to me that it would be fun to do a Probability Scarf using a different probability distribution. You could find a generator based on another distribution (where the colors wouldn’t all be evenly distributed) and use that to decide which color to use. This would be fun if you wanted to use a second or third color just for highlights. Or maybe you didn’t have the same amount of each yarn. Maybe that will be a future project….

My posts on Mathematical Knitting and related topics are now gathered at Sonderknitting.

Pascal’s Triangle Shawl #2

Tuesday, May 19th, 2015

Hooray! Hooray! Today I finished my second, prettier Pascal’s Triangle Shawl!


Pascal’s Triangle is the triangle with 1s on the edges, where each entry is the sum of the two entries above it.

So the beginning rows work like this:

1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1


Now, what I did was choose a color of yarn for each prime. Then each entry in the triangle is factored, and each number is shown by the colors of its factors.

I did the same thing with my first Pascal’s Triangle Shawl. With this one, since there are only the primes 2, 3, 5, 7, 11, and 13, I decided to use progressively darker shades of pink and purple, so the shawl would gradually get darker.

Here is a closer look at a section of the shawl:

Right Side

This next picture shows that along the second row, we have the numbers simply in sequence.

Right and Top

For math nuts, each row also contains the binomial coefficients, the coefficients in the expansion of

This means that the rth entry in the nth row can be calculated with the formula:
n!/(n-r)! (Counting the entries in each row as 0 through n.)

Some examples: The 2nd entry in the 5th row is (5×4)/(2×1) = 10

The 3rd entry in the 7th row is (7x6x5)/(3x2x1) = 35

Now, I factor all the numbers in my shawl, so for big numbers, it doesn’t matter what the actual number is, but the factorization is easy from the formula.

For example, the 4th entry in the 15th row is (15x14x13x12)/(4x3x2x1) = 3x5x7x13

You can see some of the bigger numbers in this picture:

Right Factored

Now, there are a couple of characteristics which I believe make the shawl especially beautiful.

One is that because these are the binomial coefficients, once you get to the row of a prime number, every entry in that row has the prime for a factor.

This is easier to see with the actual shawl in front of you, but here again is the big picture. You can see that once a new color starts, it goes all the way across the row.


What’s more, by the distributive law, since every entry in a prime row has that prime as a factor, all the sums of those numbers will also have the prime for a factor — and we end up having inverse triangles of each color.

Here’s some more detail:



Of course, the very coolest thing about it is that, even if you have no idea of the math involved, the combination is beautiful.

And that simply makes me happy.

Modeling Shawl

My posts on Mathematical Knitting and related topics are now gathered at Sonderknitting.