Archive for the ‘Mathematical’ Category

Review of How Much Does a Ladybug Weigh? by Alison Limentani

Saturday, March 18th, 2017

How Much Does a Ladybug Weigh?

by Alison Limentani

Boxer Books, 2016. 28 pages.
Starred Review

The more I look at this book, the more I like it. Right now, I’m planning to use it for my next Toddler and Preschool Storytimes, and even bring it to Kindergarten and first grade classes for booktalking. The idea is simple, but it’s got so much depth.

Here is the text of the first several pages:

10 ants weigh the same as 1 ladybug.

9 ladybugs weigh the same as 1 grasshopper.

8 grasshoppers weigh the same as 1 stickleback fish.

7 stickleback fish weigh the same as 1 garden snail.

You get the idea! The book progresses, counting down, through starlings, gray squirrels, rabbits, and fox cubs to 1 swan. Then, of course, to finish off, we learn:

1 swan weighs the same as 362,880 ladybugs.

The illustrations are simple and clear. This whole book could almost be thought of as an infographic, except that the animals are not icons, but detailed illustrations.

I love that the animals chosen are not your typical animal-book animals. But most of them (except maybe the stickleback fish) are ones a child is quite likely to see in their own yard or neighborhood.

The back end papers list average weights of all the animals (in a colorful diagram) with the note, “Different animals of the same species can vary in weight, just as different people do. All the weights in this book are based on animals within the average healthy weight range.”

I love the way this is a counting book, a math book (about relative weight and even multiplication), a beginning reader, and a science book (about these different species).

It’s also a beautiful picture book. The note at the front says, “The illustrations were prepared using lino cuts and litho printing with digital color.” They are set against lovely solid color backgrounds, so the animals show up nice and clear.

I have a feeling that reading this book frequently with a child will get that child noticing small animals and insects in the neighborhood and thinking about weights and differences and good things like that.

A truly brilliant choice for early math and science thinking.

boxerbooks.com

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Find this review on Sonderbooks at: www.sonderbooks.com/Childrens_Nonfiction/how_much_does_a_ladybug_weigh.html

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?

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 How to Bake Pi, by Eugenia Cheng

Tuesday, January 10th, 2017

How to Bake π

An Edible Exploration of the Mathematics of Mathematics

by Eugenia Cheng

Basic Books, 2015. 288 pages.
Starred Review
2016 Sonderbooks Stand-out: #5 Nonfiction

I have a Master’s in Math, so I love math books for a general audience. Besides, my math degree is very old by now, so a book like this taught me about a whole field of mathematics I hadn’t known about before. And it’s written by a woman!

She had me from the Prologue, where she debunks some math myths and begins with a recipe. Here are some parts I especially liked:

Cooking is about ways of putting ingredients together to make delicious food. Sometimes it’s more about the method than the ingredients, just as in the recipe for clotted cream, which only has one ingredient — the entire recipe is just a method. Math is about ways of putting ideas together to make exciting new ideas. And sometimes it’s more about the method than the “ingredients.”

Here’s about the myth that you have to be really clever to be a mathematician:

Much as I like the idea that I am very clever, the popular myth shows that people think math is hard. The little-understood truth is that the aim of math is to make things easier. Herein lies the problem — if you need to make things easier, it gives the impression that they were hard in the first place. Math is hard, but it makes hard things easier. In fact, since math is a hard thing, math also makes math easier.

Here’s talking about what it’s like to do research in math:

It’s true, you can’t just discover a new number. So what can we discover that’s new in math? In order to explain what this “new math” could possibly be about, I need to clear up some misunderstandings about what math is in the first place. Indeed, not only is math not just about numbers, but the branch of math I’m going to describe is actually not about numbers at all. It’s called Category Theory, and it can be thought of as the “mathematics of mathematics.” It’s about relationships, contexts, processes, principles, structures, cakes, custard.

Yes, even custard. Because mathematics is about drawing analogies, and I’m going to be drawing analogies with all sorts of things to explain how math works, including custard, cake, pie, pastry, donuts, bagels, mayonnaise, yogurt, lasagna, sushi.

True to her promise, she begins each chapter of her book with a recipe, and uses the recipe to illustrate the math about the recipe on the conceptual level.

Abstract Algebra was always one of my favorite fields of math, and Category Theory is a level of abstraction higher. What could be cooler than that?

But if the idea of extreme abstraction doesn’t get you as excited as it does me, think of it as math concepts explained through recipes. That conveys better how friendly this book makes the concepts.

She has analogies for almost everything. Here’s where she explains what abstraction is:

Abstraction is like preparing to cook something and putting away the equipment and ingredients that you don’t need for this recipe, so that your kitchen is less cluttered. It is the process of putting away the ideas you don’t need for the present purposes, so that your brain is less cluttered.

Here’s her explanation of proof by contradiction:

Imagine trying to “prove” that you really need to boil water to make tea. You would probably just try to make tea without boiling the water. You discover that it tastes disgusting (or has no taste at all) and conclude that yes, you do need to boil water to make tea. Or you might try to “prove” that you need gas to make your car go. You try running it on an empty tank and discover it doesn’t go anywhere. So yes, you do need gas to make your car go.

In math, this is called proof by contradiction — you do the opposite of what you’re trying to prove, and show that something would go horribly wrong in that case, so you conclude that you were right all along.

I think this book is truly beautiful. And I suspect it might provide glimmers to people who have never before seen beauty in math at all. If that’s not enough to appeal to potential readers, well, it has recipes.

basicbooks.com

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Find this review on Sonderbooks at: www.sonderbooks.com/Nonfiction/how_to_bake_pi.html

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?

Review of What in the World? Numbers in Nature, by Nancy Raines Day and Kurt Cyrus

Friday, September 9th, 2016

what_in_the_world_largeWhat in the World?

Numbers in Nature

by Nancy Raines Day
illustrated by Kurt Cyrus

Beach Lane Books, New York, 2015. 32 pages.

This is a simple picture book introducing a little bit of counting and a little bit of science to young readers.

Each number is introduced by a question, “What in the world comes . . .” and gentle pictures by the seaside illustrate each set.

Here’s an example from the middle:

What in the world comes four by four?

Petals of poppies, hooves – and more.

What in the world comes five by five?

The arms of sea stars, all alive.

There are only two lines per double-page spread, and plenty of open space in each painting, so this is for young readers who can handle the gentle pace. It would make a nice bedtime book, since the book finishes up with “sets too big to count.” The final two spreads show us a darkening sky with the words

Stars in the sky –

a vast amount!

You can hear from these examples that the rhyming isn’t stellar, but it’s doesn’t quite cross the line into bad. One other quibble I have is that on the sets of ten page with “Fingers and toes that wiggle and bend,” the picture of the boy does show his fingers and toes (in the water), but his arms are crossed with one thumb hidden – so you can’t use the picture to count ten fingers and ten toes.

However the simple idea – a counting book based on nature – is a lovely one. This is a gentle book that naturally leads into counting with children things in the world around them. A great way to practice counting and a great way to open their eyes to nature.

NancyRainesDay.com
KurtCyrus.com
KIDS.SimonandSchuster.com

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Find this review on Sonderbooks at: www.sonderbooks.com/Childrens_Nonfiction/what_in_the_world.html

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.

normal_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.

Normal_Colors

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.

Normal_Colors2

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?]

Normal_Colors3

In this version, the lighter colors were more prominent.

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

normal_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:

normal_detail

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

normal_detail2

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!

normal_detail3

Review of Butterfly Counting, by Jerry Pallotta

Saturday, July 2nd, 2016

butterfly_counting_largeButterfly Counting

by Jerry Pallotta
and Shennen Bersani

Charlesbridge, 2015. 32 pages.
Starred Review

I’ll admit, I am already a huge Jerry Pallotta fan. Why? Because 27 years ago, The Bird Alphabet Book was one of the very first books my child loved. We read it so often, she could recite whole paragraphs from the book with her cute toddler voice. Phooey, 27 years later, I can recite whole paragraphs from the book. (I especially remember, “Wait a minute, bats are not birds! Although they have wings and can fly, bats are mammals…. Get out of this book, you bats!”)

This book does a little of that playing with the reader as well. It starts with a spread of 20 moths. After counting them,

But wait . . . these are not butterflies! These are all moths. We tricked you! Moths can be very colorful.

Then it goes on to count butterflies of different varieties. The first ten butterflies are red, blue, green, purple, orange, black, white, pink, yellow, and brown. The next nine are multicolored and patterned butterflies. Then for 20 to 25, they look at the lifecycle of the butterfly, beginning with twenty Pipevine Swallowtail butterfly eggs.

Each page tells us the word for butterfly in another language. And the book is full of facts about the different varieties of butterflies.

And the book is so beautiful! The illustrator has made stunning paintings of each variety of butterfly (or moth).

It’s so easy for me to imagine a small child, like young Jade, avidly learning and reciting these facts.

The last page shows a lovely creature with wings that go from yellow to bright pink.

A butterfly in Great Britain is called a butterfly. But don’t be silly! This is not a butterfly. It is a grasshopper. Should we write a grasshopper book next?

jerrypallotta.com
shennenbersani.com
charlesbridge.com

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Find this review on Sonderbooks at: www.sonderbooks.com/Childrens_Nonfiction/butterfly_counting.html

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?

Review of The New York Times Book of Mathematics, edited by Gina Kolata

Wednesday, June 1st, 2016

ny_times_book_of_mathematics_largeThe New York Times Book of Mathematics

More than 100 Years of Writing by the Numbers

edited by Gina Kolata
Foreword by Paul Hoffman

Sterling, New York, 2013. 480 pages.

If you’re at all interested in mathematics, this is a fascinating book covering the history of major developments in mathematics in the twentieth century and the start of the twenty-first, as told in the pages of The New York Times.

Since the articles are from The New York Times, they are written for the general public, and the articles aren’t too lengthy. I mostly read one a day for a very long time. Incredible as it may seem, the book was never on hold when I wanted to renew it.

My only complaint was that I had to learn to check the date – given at the end of each article – before reading the article rather than after, because all are reported as happening in the present – it was nice to know when the major development had actually happened in 1936 (though there are more from recent years than going that far back).

Of course, having once been in a PhD program in Mathematics (though I settled for my Master’s), I was extra interested to get tastes of what’s going on before and after my time in the math department. For me, it was interesting to place the articles as before or after my time at UCLA.

The articles are grouped in chapters of related articles. You’ll get an idea of what to expect from the chapter titles: “What Is Mathematics?”; “Statistics, Coincidences and Surprising Facts”; “Famous Problems, Solved and As Yet Unsolved”; “Chaos, Catastrophe and Randomness”; “Cryptography and the Emergence of Truly Unbreakable Codes”; “Computers Enter the World of Mathematics”; and “Mathematicians and Their World.” Try this book for a bird’s-eye view of that fascinating world.

sterlingpublishing.com

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Find this review on Sonderbooks at: www.sonderbooks.com/Nonfiction/ny_times_book_of_mathematics.html

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?

Prime Factorization Coloring Sheets

Wednesday, April 27th, 2016

I’ve posted several Prime Factorization Coloring Sheets on my Sonderknitting page lately.

I decided I should try coloring them myself, so I could post a thumbnail of each one. I had a lot of fun doing it, and was reminded of lots of cool properties I discovered from knitting my prime factorization sweater and looking at these charts.

I have a manuscript for a math-related children’s nonfiction book about using math to make codes with colors. Originally, I put several of these charts into the book — but I eventually decided it was a distraction and decided to put them on my website instead.

But they show all sorts of cool things!

First, there’s the ten-by-ten prime factorization chart using ordinary, decimal numbers.

pf_chart_hand_colored_for_blog

Coloring this chart gives you a great feeling for factorization and multiples. I posted about watching a second grader color it. I think of it as more for older kids, who are learning about primes and multiples, or indeed adults, in keeping with the adult coloring book craze. But watching a second grader color it assured me that it can give insights to anyone. (I made the instructions such that you don’t even have to know how to multiply. Just color every second square the color for 2, every third square the color for 3, and so on.)

Now, in my original sweater, I put rows of 8 on the back and rows of 2 and rows of 3 on the sleeves. The prime factorization charts in different bases are the same idea.

First, they give you a feeling for how different bases work.

Here’s the sheet for octal, base 8:

octal_chart_hand_colored_for_blog

You can color it exactly the same way as you did the ten-by-ten chart. Color every second square with the color for 2, every third with the color for 3, and so on. If you take the time to do that, you’ll grasp how the numbers count up to 7 and then use the next digit, since place value in octal gives the ones digit, the eights digit, and the sixty-fours digit.

The chart also makes a good way to translate between octal and decimal. (Though you can just multiply the eights digit times eight and add the ones digit.)

But I enjoy some of the other patterns.

The first, most obvious pattern is that in the decimal chart, the multiples of 5 and the multiples of 2 line up vertically (as well as the multiples of 10, which are both). That’s because 10 = 2 x 5.

In the octal chart, the multiples of 2 line up vertically, since 8 = 2 x 2 x 2. So do the multiples of 4 — each with two factors of 2, and the multiples of 8 — each with three factors of 2.

In the Base 6 chart, as you’d expect, the multiples of 2 and the multiples of 3 line up vertically. (And the multiples of 6, with a factor of 2 and a factor of 3, do as well.)

base_six_hand_colored_blog_size

But it’s also fun what happens to the color for Base Plus One and Base Minus One.

In the 10×10 chart, look at what happens to the color for 11, orange, and the multiples of 11. They go diagonally to the right up the chart: 11, 22, 33, 44, . . .

In the 10×10 chart, 9 is represented by two sections of blue, for 3 x 3. These colors go diagonally up the chart in the opposite direction: 9, 18, 27, 36, . . .

In the 8×8 chart, the octal number 11 is the decimal number 9 — so it is still represented by two sections of blue. But since 9 is one bigger than our base in that chart, the two sections of blue go diagonally up the chart to the right — just like 11 in the decimal chart.

In the octal chart, the color for 7, purple, goes diagonally up the chart to the left, with the octal numbers 7, 16, 25, 34, . . . .

In the 6×6 chart, we’ve got the same patterns, this time with 7 (which is 11 in base six) and 5.

7 (purple) goes diagonally right up the chart, and 5 goes diagonally left up the chart.

And we’ve got the same patterns in a 7×7 Base Seven chart:

base_seven_hand_colored

Notice that since 7 is prime, no colors line up except purple, the color for 7.

And the colors for 8 and 6 go diagonally up the chart.

The Hexadecimal chart in base 16 is even more interesting:

hex_hand_colored_blog_size

Notice how all the multiples of 2 line up vertically, with multiples of 4, 8, and 16 also lined up.

11 in Base 16 is decimal 17, which is brown, and it acts like all the other 11s, going diagonally up and to the right.

1 less than 16 is F = 15, and the blue and green colors for F go diagonally up and to the left.

Before I finish I want to mention one more pattern I noticed from looking at these charts. It’s the familiar trick in Base 10 of the rule for figuring out if any number is a multiple of 9: Just add up the digits, and they will be a multiple of 9.

The reason this works is that 10 is congruent to 1 mod 9.
In base 10, each decimal place represents a number multiplied by a power of 10.
In base 9, that’s going to be the same as multiplying by 1 — so if you add up the digits, you get what the number is congruent to mod 9.

If none of that made any sense to you, just know this:
If you add up the digits of a base 10 number (and if you get a number bigger than 9, add them up again), your result is the remainder you’ll get if you divide the number by 9.

Since multiples of 9 have no remainder when divided by 9 — the digits of multiples of 9 in base 10 always add up to multiples of 9. (And by the same reasoning, the digits of multiples of 3 in base 10 always add up to multiples of 3.)

But you might have noticed when looking at the diagonal colors:

In Base 8, the digits of multiples of 7 always add up to multiples of 7.

In Base 6, the digits of multiples of 5 always add up to multiples of 5.

In Base 7, the digits of multiples of 6 always add up to multiples of 6.
And the digits of multiples of 2 always add up to multiples of 2.
And the digits of multiples of 3 always add up to multiples of 3.
(Use the colors to tell which numbers these are in Base 7.)

In Base 16, the digits of multiples of F (15) always add up to multiples of F.
And the digits of multiples of 5 always add up to multiples of 5.
And the digits of multiples of 3 always add up to multiples of 3.
(Use the colors to tell which numbers these are in Base 16.)

Forgive me, but I think these patterns are Awesome!

Let’s face it, you’ll see them much more clearly if you color the charts yourself!

Download the coloring charts at Sonderknitting!

Happy Coloring!

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.

coded_affirmations_scarf

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

Normal Distribution Coloring Sheets

Wednesday, February 24th, 2016

I’ve made a Normal Distribution Coloring Sheet and posted it in my Mathematical Knitting Gallery, Sonderknitting.

I thought it would be fun to talk more about it and show some examples.

The reason it’s in my Mathematical Knitting Gallery is that the idea began with knitting.

First, it was my Probability Scarf. I read this idea somewhere. Just choose six colors that look good together. Knit the scarf lengthwise. Assign the numbers 1 through 6 to the six colors. For each row, roll a die to decide which color to use on that row. Flip a coin to decide whether to knit or purl.

Here’s how that scarf came out:

Probability_Scarf

But in this scarf, all the colors are equally likely. This is called a uniform distribution. What if the colors were chosen from a normal distribution, a bell-shaped curve? That’s what I did with Jade’s Outliers Scarf, using bright colors for the outliers, plainer colors for the middle of the curve.

OutliersScarf

But then I thought it would be fun — and much, much quicker — to do this with colored pencils or crayons. So I made a coloring sheet that is just a grid. But the instructions explain how to use random numbers chosen from a normal distribution to color the sections in the grid.

The scarf used three colors, plus a rainbow yarn for the outliers. I decided to use four shades of colored pencils: dark blue for within half a standard deviation of the mean, dark purple for between one-half and one standard deviation, green for one to one and a half standard deviations, and light blue for one and a half to two standard deviations from the mean. Then I used a red marker for the outliers more than 2 standard deviations out from the mean. (I may try this in a scarf, so it was nice to check how it looks first.)

Here’s how it turned out:

normal_distribution_hand_colored_small

Since a lot of characteristics in people or in nature have a normal distribution, this gives a good feel for how people vary. It also explains why the outliers might feel like oddballs. And why one outlier might have a hard time finding another like themselves. But don’t change, outliers! You are what makes life beautiful!

I’m still going to try some other color schemes. I’m thinking it might be time to buy some colored pencils with more shades.

But meanwhile, it occurred to me that I could get more shades if I used computer coloring.

My grid is a table in Microsoft Word. And you have the option of coloring each cell, specifying a number between 0 and 255 for the red, green, and blue elements in RGB mode.

So I went back to random.org and generated numbers from a normal distribution with 128 (right in the middle) as the mean and 42 as the standard deviation. So the only way the numbers would go past 0 or 255 would be more than 3 standard deviations out from the mean. (With 990 numbers generated, only one did.) I’m thinking about doing it again using a standard deviation of 64, in which case there would be more variation, and you’d have more using 0 or 255.

It was interesting to do. The majority turned out to be grayish. You’d get the brightest squares when one element was very different from the other two.

colored_normal_distribution

It took a long time — I’m sure it would be fairly simple to create a program that would generate one of these charts, so maybe I’ll do that sometime in the future. I’m also thinking about doing the same thing but using the HSL color model available in Word. HSL stands for Hue, Saturation, and Lightness — but it also uses numbers 0 to 255 for each one.

Meanwhile, I feel like my intuitive grasp of the normal distribution has grown.

But mostly, I think these are pretty.