Image by PublicDomainPictures from Pixabay
What makes something a maze?
Google Dictionary says that a maze is “a network of paths and hedges designed as a puzzle through which one has to find a way.” Similarly, a labyrinth is “a complicated irregular network of passages or paths in which it is difficult to find one’s way; a maze.” It’s a place meant to confuse you, a place where you’re meant to get lost. And the odd thing about it is, it’s fun. Have you ever been in a corn maze? How long did it take you to find your way out?
Mazes show up in mythology and folklore a lot. The most famous one is the labyrinth created by Daedalus for the Minotaur on the Greek island of Crete. Mazes also show up throughout the world, including Scandinavia.
They’re made of stone or hedges or just mounds of earth. They’re cut into crops or turf. They’re carved on stone, painted on walls, inlaid in floor mosaics, minted on coins, woven into cloth. Why are we so fascinated by mazes?
Well, if you’ve ever gotten lost in a store, or been frustrated driving through a city with lots of one-way streets, or lost your bearings while hiking in the woods, I’m sure you found yourself wishing there was an easy way to find your way out, short of scattering breadcrumbs like Hansel and Gretel.
The nice thing about mazes is that they follow rules, and if you know the rules, you can use them to get out. All of the mazes shown above can be solved using the right-hand rule (which is also a rule that you use in physics in a very different context). As you enter the maze, put your right hand on the wall. Keep following the wall. Since those mazes are one continuous line, you will eventually end up back at the entrance. This works with just about any maze that has an outer boundary, including corn mazes. It’s the long way around, certainly, since you’ll end up walking every single pathway in the maze, but it’ll get you there.
But what if the rules are a little different? In An Elf’s Equations, I have three mazes. The first, in the Sylvan Vale, is a maze that you can’t solve without knowing the Fibonacci sequence (or someone tells you the sequence of steps to take). The second maze, around the World Tree in Vanaheim, is a very logical, squared-off maze full of right angles and dead ends. The right-hand rule would work just fine in there, but it’s straightforward enough that you’d likely be able to walk through it easily.
And then there’s the maze of Pthagor, the math dragon. It looks so simple at first.
You could just walk right through this, right? Ah, but there’s a catch, as Sagara and her friends discover:
At the mouth of the cavern, they found a sign printed in several different languages, including Vanir runes, Dragon script, Canto, English, and several more they did not recognize. It read:
To go forward, turn only to the right.
“What does that mean?” Petunia asked. “If it’s a riddle, I don’t get it.”
“I think it’s instructions,” Max replied.
Cretacia snorted. “And what if we don’t go right?”
“One way to find out,” Sagara said, and she strode into the cavern.
Before her, laid out on the smooth floor, a maze had been marked out in faintly glowing paint. At first, she thought it was a simple square maze, like the one in the Vanir garden of the World Tree, but then she looked again. From the entrance, paths went left, straight and right. They branched and turned. About fifty feet away, Sagara could see the exit, leading deeper into the cavern.
“Ha!” Cretacia yelled. “Easy!” She ran straight into the maze, heading for the exit, but when she crossed one of the painted lines, she disappeared.
“Cretacia!” Millie squealed.
“Great horny toad warts!” Cretacia said behind her. “That was really weird. One second, I was in there, and the next second, I was back out here. I wonder what happens if I turn left?”
“No, wait,” Sagara said, but Cretacia had already run into the maze and took the first left turn. Again, she disappeared.
“Hey!” came her voice from above. “How do I get down from here?”
Sagara stepped back from the cavern mouth and found Cretacia sitting on the bulbous tip of the troll’s nose.
— pp. 233-4
I based this maze on one I encountered outside the Tekniska Museet in Stockholm. It’s a tricky maze because it has different rules. Can you find your way through it? The solution to Pthagor’s maze is in An Elf’s Equations.
Now here’s the activity: what other rules can you change to make a maze more tricky? Maybe you can only turn if you can’t go straight any more, or you have to turn if you can, or you have to jump over things. Try making a maze of your own with different rules! Can you make something that looks simple but is actually hard to get through? If you do, take a picture and share it with me either here or on my Facebook page. I’d love to see your creations!
And if you’re wondering what that troll’s nose looked like, here’s a troll I found under a bridge in Seattle (well after I’d written this scene!) who was friendly and obliged me when I asked for a selfie with him.
[Note: this is a longish post. If you want to skip straight to the activity, scroll down to the bottom and look for the runes.]
I was stuck.
I’d turned in A Pixie’s Promise to my publisher in May 2018 and had intended to finish up An Elf’s Equations by August 2018, but it was July already, and I’d barely written anything. I had thought this would be easy. After all, it was already half-written.
An Elf’s Equations was originally the second half of A Pixie’s Promise, but the fabulous Corie Weaver at Dreaming Robot Press thought that there was entirely too much going on in A Pixie’s Promise and, furthermore, that for the second half of the book, Sagara was really the focal character. She wanted me to split the two plotlines into two books and make Sagara the protagonist. After much groaning and complaining, I agreed, and it took me only three months to expand and reshape A Pixie’s Promise into a much more fully realized story focused firmly on Petunia. So I expected that it would take me about the same amount of time to finish up An Elf’s Equations. Right?
Wrong. Terribly, terribly wrong. First, I couldn’t just change who was speaking from Petunia to Sagara, because Sagara wouldn’t say the same things Petunia does (far fewer bad jokes, much more sarcasm). And I couldn’t just have Sagara do the things I’d had Petunia do, not least because Sagara isn’t six inches tall. She’d make different decisions, come up with different solutions. Second, I hadn’t done all the character work that I’d done for Petunia. I had a vague idea of her background, I knew that her mother had been exiled to the Logical Realm, and that Sagara really didn’t get along with her grandmother. And I knew that Sagara was fascinated by math. But why? What drove Sagara? And how would she end up leading the expedition to Vanaheim to save her friend Thea?
The more I tried to just gloss over the differences, the more the story unraveled on me, until I was utterly lost and deeply despondent that I would never get the novel done.
And then, something magical happened.
My family and I went to Sweden and Finland to visit my husband’s family and give our daughters some sense of their cultural heritage. I’m also 1/16th Swedish, thanks to a great-great-grandmother who emigrated from Malmö. When we arrived in Gothenburg in the middle of July, we went directly to an island on Lake Mjörn. Sweden was experiencing a terrible drought that summer. Forest fires raged in the north of the country, though quite far from us. The water level in the lake had dropped so much that we had to move the dock to keep the motorboats and sailboat from scraping bottom. And the leaves on the birch trees were turning yellow and dropping everywhere.
One night, around 3 AM local time, I awoke, jet-lagged and disoriented and needing to use the outhouse. A single solar panel provides all the electricity for the island, and that meant no power at night. I had a flashlight, but there was a beautiful full moon. It was easy to find my way from our little cottage to the outhouse because the path there was strewn with yellow leaves, which practically glowed in the moonlight. And as I walked, I imagined Sagara, riding her deer on the Path in the Enchanted Forest, following her grandmother home after the trial of Cretacia. I imagined all the emotion boiling up within her as she herself remembered running through the Sylvan Vale in the moonlight to the trial of her mother. And I started to hear her talking to her grandmother, angrily, blaming, and so very hurt and lonely.
I grabbed my laptop and wrote until dawn was beginning to lighten up the horizon and my laptop’s battery had completely run out. I’d just banged out the whole of chapter two, and I went back to my bed with a smile on my face.
Scandinavia inspired me in many other ways during that trip. Here’s a photo album I put together of some of the things that found their way into An Elf’s Equations.
One of the things that had fascinated me on previous trips as well as this one were the runestones we often came across. Viking runes are an ancient writing system, an alphabet made entirely of trees. They were used throughout Scandinavia and also show up in the British Isles and Ireland. Robert Graves, who was a historian as well as the author of Count Belisarius and Hercules, My Shipmate, wrote about the fabled battle of the trees. Taken literally, it sounds as if two bards used magic to call up entire forests to do battle. In fact, it was a metaphor for a poetry competition, and the trees doing battle were runes upon parchment. Each rune represents a tree as well as a single sound.
Here are a few resources where you can learn more about Viking runes and runestones:
In preparation for An Elf’s Equations, I modified the standard runic alphabet for the Vanir. Vanaheim, the Realm that Sagara and her friends travel to, is based upon a scrap of Norse mythology in which the universe is composed of nine worlds or realms, all connected by the World Tree. We live in Midgard, the Norse gods live in Asgard (as you’ve seen in the Avengers movies). Those gods were called the Aesir, but they had cousins in another realm, the Vanir, who lived in the realm of Vanaheim. All that remains of the myths about the Vanir is that they specialized in nature magic, which I interpreted to mean elemental magic, and that the goddess Freya/Freja/Frigga was originally a Vanir.
Cretacia is half Vanir, which is one of the reasons she’s placed under geas to retrieve Thea from Vanaheim at the end of A Pixie’s Promise. Her father is a high-ranking Vanir and she speaks the language. Cretacia also happens to be dyslexic, which means she has some trouble reading and thinks she’s terribly stupid. But her father Ljot (that’s pronounced Lee-OHT) explains to her that all Vanir have trouble reading, which is why they invented a magical language that is easy to read.
I had heard about fonts that are easier for dyslexics to read. There’s some debate about whether this actually works, but I have a friend who swears by it. Whether it works or not, I was intrigued by the idea, and I thought, well, if you have magic, couldn’t you make an alphabet that dyslexics can read easily? So I modified the runic alphabet, making all the runes look the same forwards and backwards — no mistaking “b” for “d” in this language. Here’s what I came up with:
Can you find the runes to open the Vanaheim portal on the cover of An Elf’s Equations?
For those of you who enjoyed creating secret messages last week, you can now write secret message using my runic alphabet.Note that there are no runes for Q or V. In many Northern European languages, W and V are almost interchangeable. And for Q, I recommend using CW together. The Vanir Runes also have runes for two sounds that are represented by multiple letters in English: NG and TH. And the rune for Magic has no corresponding letter. That’s because it’s not something that’s spoken aloud, it just indicates the need to use magic.
Now that you’ve seen how easy it is to create an alphabet, why don’t you try creating one yourself? If you do, take a photo and share it in the comments here or on my Facebook page. Enjoy!
One of the fun things you can do with numbers is create codes. Codes are ways of organizing information in different, useful ways. In one sense, the Fibonacci sequence is a code that nature uses to create the best spacing between things. If you flip over An Elf’s Equations, you’ll find a barcode on the back. Bookstores use barcodes as a quick and easy way for their computers to identify books. And codes can be used to keep information secret and safe.
But a code doesn’t have to be very complicated to work well. Here’s a very simple code that borrows its complexity from other objects. If you and a friend want to send each other secret messages, all you need is two copies of the same edition of a book. Write your message, then look through the book for all the words of the message. Then write down just three numbers to encode the message: the page number, the line number, and the number of the word within that line. For example, the word “Cretacia” appears on page 75, line 5, word 6. Go check it out!
This kind of code is called a book cipher. To make the code even harder to crack, people write the location as just the three numbers. So “Cretacia” above would be (75, 5, 6). The word “Author” appears on page 311, line 1, word 3, or (311, 1, 3).
Got it? Okay, then I have a secret message for you, and you can only decode it if you have a copy of An Elf’s Equations. Ready?
(74, 4, 7) (151, 21, 2) (261, 12, 2) (108, 11, 4) (245, 11, 8): (48, 28, 6) (48, 9, 4) (176, 14, 4) (61, 20, 7) (79, 4, 1) (63, 23, 9) (212, 17, 1) (287, 13, 5) (207, 7, 8) (214, 4, 5)? “(56, 24, 1) (65, 7, 8).”
Did you figure it out? Don’t post it in the comments! Let other people figure it out for themselves. However, feel free to create your own secret messages and post those, either here or on my Facebook post about secret messages. And you can use this method to send secret messages to your friends, as long as you each have an identical copy of the same book. I recommend that you choose a book with a broad vocabulary. That way, you’ll have lots of different words to choose from.
One thing I’m curious about. I actually don’t have an eBook copy of An Elf’s Equations, so I don’t know if the pages, lines, and words fall in the same places. Could someone with an eBook version check? If it turns out to be complete gibberish, go ahead and post it in the comments, just to show how horribly wrong a code can go!
Tomorrow, I’ll be doing a live question and answer session. It’s much easier for me to answer questions if I have them in advance, so please post your questions in the comments either here or on Facebook, and I’ll do my best to answer them. Thanks!
Do you know where chocolate comes from? Do you really?
You might have heard that chocolate comes from cacao beans, and that these grow on cacao trees, but did you know that:
* The scientific name for cacao trees is Theobroma cacao? That’s where Thea gets her name.
* There are four varieties of cacao tree: forestero, criollo, trinitario, and nacional. The nacional variety was found in Peru just nine years ago!
* The genus Theobroma is probably millions of years old, but modern cacao trees first appeared around 10,000 years ago in the Amazon basin, and they’ve been cultivated for 3,000 to 5,000 years.
* Cacao trees only grow in tropical climates very close to the Equator. Most cacao trees grow in West Africa today, but they are also grown in Central and South America, Indonesia, the Caribbean, and Hawaii. So unless you live in one of those places, you can’t have a cacao tree in your backyard.
* Cacao trees can grow to be 30 feet tall, even if they grow in the shade of much larger trees.* Cacao trees can bloom and bear fruit all year around, but they are usually harvested twice per year.
* A single cacao tree can have as many as 6,000 blossoms, but not all of them turn into fruit.
* Cacao fruit, called pods, can be many different colors, from bright yellow to green, brown, red, or even purple!
* Cacao pods grow on the trunks of cacao trees.
* Cacao trees are very vulnerable to fungal infection, and cacao habitats are slowly shrinking due to deforestation and climate change.
* Chocolate was first brought to Europe in 1502, but it wasn’t made into solid bars until 1848.
* The first chocolate house in the United States in 1765, owned by John Hannon and Dr. James Baker, after whom Baker’s Chocolate is named.
* When I was a student at MIT in Cambridge, MA, I used to walk past a chocolate factory every day on my way to class.
If you’re interested, you can read more about cacao trees here:
I’m fascinated by cacao and chocolate, which is why one of my characters is Thea, a baby magical intelligent cacao tree.
Thea was no ordinary tree. The Enchanted Forest School that Petunia and Millie attended took place in the branches of an enormous, intelligent, and magical oak tree, a dodonos named Quercius. Dodonoi were extremely rare — there were only four in the whole Enchanted Forest Realm before Thea came along. …At school, Millie had accidentally transformed a green bean into a cacao bean, and because it had sprouted so close to Quercius, it became a dodonas, an intelligent, magical tree: Thea. One day, Thea would walk and talk and even attend school, learning to use her magic. But for now, she was still a baby and needed care and protection. — A Pixie’s Promise, p. 17.
Now you can create your very own Thea to care for and protect! You will need:
- Paper and/or cardstock
- Cardboard from a cereal box or shipping box
- Tape and/or glue
- Markers, crayons, colored pencils, and/or paint
- Optional: pipe cleaners, popsicle sticks, stickers, glitter, googly eyes – whatever is fun for you!
- Download MakeYourOwnThea-template and print it out. You might want to print extra copies of page 3. If you can print it on cardstock, you can skip steps 3 and 4 below.
- Cut out the two tree shapes. Be sure to cut the slot in each trunk. You might want to ask an adult for help with this.
- Use the tree shapes as stencils and draw their outlines onto some cardboard.
- Cut out the cardboard tree shapes. You might want to ask an adult for help with this. Adults, for heavy cardboard, you might want to use a box cutter or other sharp implement.
- Slide the two tree shapes to form an X. The tree should be able to stand on its own.
- Color the leaves, cacao pods, and flowers on page 3.
- Cut them out and decorate the tree’s branches and trunk with them. While Thea is actually too young to have flowers or fruit (she’s less than a year old, and cacao trees usually don’t bloom until they’re 3-5 years old), you can use them if you want.
- Add other decorations, such as eyes, pipe cleaners, and stickers. Make your tree magical!
- Take a photo and post it in the comments on the Dianna Sanchez Facebook page.
Tomorrow, I’ll be posting a secret message! Have your copy of An Elf’s Equations handy to decode it.
Also, on Friday, I’ll be doing a live Question and Answer session. If you post questions in the comments on my Facebook page, I’ll gather them up and answer them.
Image by Karin Henseler from Pixabay
I have long been fascinated by the Fibonacci sequence because it seems to show up everywhere. You might have noticed it, that certain swirl that shows up on things. Here are a few examples:
In An Elf’s Equations, Sagara describes it like this:
Start with 1, then add the previous number to it (in that case, 0), then add the previous number to that number… 1, 1, 2, 3, 5, 8, 13, 21… Sagara could describe this elegantly as an equation: xn= xn-1+ xn-2. Not that anyone else she knew appreciated that elegance. She could draw this as a series of blocks, then draw a line through their intersections to make a spiral, the same spiral on snail shells and flower petals and pine cones, even faces. Taking the ratio between any two consecutive numbers in the sequence and averaging them gave Sagara the Golden Ratio, 1.618034, which could usually be simplified to 1.6. This number appears over and over and over again in nature. Sagara’s mother told her it even determined the shape of great storms and swirls of stars in the sky. –pp. 46-47
I think Sagara does an okay job of describing the Fibonacci sequence, but I know someone else who does a spectacular job of explaining it: Vi Hart. Her Doodling in Math Class YouTube series shows just how much fun math can be, and how much it influences art. She has three videos on the Fibonacci sequence. The first is the simplest, Part 2 gets a little more complicated, and Part 3 is very complicated, so see how much you can challenge yourself! I recommend you watch them several times because Vi Hart talks REALLY fast.
Want to find some spirals yourself? Ordinarily, I’d suggest that you go shopping in the grocery aisle of your supermarket and look for pineapples, artichokes, cauliflower, and stalks of brussels sprouts to try out, but since we’re all avoiding grocery stores, go outside! Pine cones, leaves on stalks, flowers (if they’re up and growing in your part of the world), and even acorn caps all have Fibonacci spirals on them. Can you find them? Can you find other spirally things in your yard?
And if you can’t, try drawing slug cats and flowers and pine cones, or come up with new creatures made out of spirals, Sagara, for example, draws spiral dragons.
Tomorrow, I’ll give you a template so you can build your own model of Thea!
Hello, everyone! I hope you’re all finding interesting and fun things to do while practicing social distancing. This week, I’m going to be posting activities for you to do while you’re home. Today’s theme is Sagara’s bedroom, which is decorated with MATH. Here’s an excerpt from An Elf’s Equations:
Sagara turned and ran lightly along a branch as broad as the Path by Millie’s house until she reached her bedroom. Pulling aside a curtain of cultivated ivy, Sagara touched a smooth river stone set in a niche beside the door, and it began to glow with soft, golden light. She entered a room made entirely of living woven branches, much like the classrooms in Master Quercius’s branches at the Enchanted Forest School. This room was much smaller and private, decorated with mathematical constructs: fractal patterns, the Fibonacci spiral, graphs and diagrams. A long strip of paper circled the leafy ceiling with the first two hundred digits of pi. — p. 31
What the heck is all this stuff? Well, here are some brief explanations of what they are and why Sagara thinks they’re cool.
Pi Goes On Forever
Pi is a strange and unusual number. It’s the number we’ve discovered that describes round things: circles, balls, even the orbits of planets (if the orbits were perfect). Draw a circle and put a dot in the exact center of the circle, then draw a line straight through that dot from one side of the circle to the other. We call that line the diameter. Now put a dot anywhere on the circle. The length of the circle if you go all the way around it is called the circumference. So if you have a piece of string, and you make a circle out of it, the length of the string is the circumference. If you measure the circumference and the diameter very carefully, you can find pi.
For example, let’s say you have a string that’s a foot long. If you make a circle out of it and measure its diameter, you’ll find that it’s a little under four inches, about 3.8 inches. Now, divide 12 inches by 3.8 inches. You can use a calculator, it’s okay. You’ll get a number with a lot of digits after it: 3.157847368 is what my calculator says. That’s pretty close to pi, but not exact because we didn’t measure the radius exactly enough. If you measure it really, really carefully, it’s more like 3.14159. Here’s a little video of how that works:
Now here’s the really cool thing. If you could measure the diameter and the circumference absolutely perfectly, the digits would go on forever. What’s more, they never repeat. Mathematicians call this a transcendental number. They have calculated millions of digits of pi, and they still haven’t found the end. Sagara loves this. She loves thinking about the fact that it goes on and on and on forever. So she glued together pieces of paper and copied out as many digits as she could and put them up all around her room. You can do that, too! How many digits of pi can you write out?
A Mobius Strip has Only One Side
Take a piece of paper. Cut it into three strips and tape or glue them into one long strip. Now give the strip one twist and tape the ends together. Here are some good step-by-step instructions. Now, put a dot in the middle of the strip, anywhere, and start drawing a line down the middle of the strip. Keep going. Eventually, you’ll reach the dot again. HOW IS THAT POSSIBLE???
It’s possible because a Mobius strip actually only has one side. That twist you put in the loop means that you connected on side of the paper to the other, making an infinite loop. Sagara thinks this is better than magic, and she has little mobius strips dangling from the branches of her room. When she gets bored, she takes one down, draws that line, and then cuts along the line to see what happens. Try it! I guarantee you’ll be surprised.
Fractals Can Be Infinite
Fractals are repeating patterns that appear in nature. Be warned! Once you start seeing them, you can’t stop.
Here’s a simple example of a fractal. Draw an equilateral triangle – that’s a triangle whose sides are all the same length and whose angles are all the same 60 degrees. Now, draw another, upside down triangle inside that triangle, with each of its points at the middle of the larger triangle’s sides. Suddenly, you have four triangles. Do that again. And again. And again.
via the Boston University Math page on Serpinski Triangles
If you had a big enough triangle, or a good enough magnifying glass, you could keep on doing this forever. The triangle doesn’t even have to be equilateral. Try drawing some triangles of different sizes and see if you can keep dividing them up the same way.
Tangrams: Many Pictures from the Same Shapes
Sagara loves turning things around and seeing them in new and surprising ways. Tangrams are images that you make from a small set of shapes. By courtesy of my friend Rebecca Rapoport, co-author of Math Games Lab for Kids, you can download and print the basic tangrams set and many different tangrams shapes to make (there are some other cool math downloads on that page, too). I recommend coloring the pieces for the tangrams many different colors, then seeing how the colors fit into the shapes. Sagara enjoys trying to create new shapes for herself and has several unique creations posted around her room.
As for Fibonacci spirals, we’ll get to that tomorrow. Happy mathing!