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This was inspired by The Phantom Tollbooth by Norton Juster and Flatland by Edwin Abbot Abbot.

When she awoke she was in a strange land. She looked around. The landscape she was in was very colorful, but quite flat. There were no hills or walls of any kind. Also, in every direction she could see there were figures. Lines, squares, triangles, dodecahedra, as far as the eye could see, each a different color. No, that wasn’t quite right. Each kind of figure was a different general color, and then each one within that subset was a shade of that color. All of the triangles were green, the squares red, the hexagons blue, and yet no color was ever repeated. She groaned. “Not math again!”

“But whatever is wrong with math my dear girl?” a chap from behind inquired. She turned around to find a very large tetrahedron staring her down. He was standing on one of his planes, and just a tad below the tip part of the triangle, where the two lines come together in a point, was a set of perfectly normal brown eyes. Beneath them were a nose and a mouth, just like any decent sort of fellow. Joan, seeing that a tetrahedron had talked to her, was instantly relieved. “Aha! I must still be sleeping. Which means I can wake up.”

“I don’t think it would be wise to wake yourself up, if you must assume that this is a dream.” The tetrahedron pleasantly smiled and continued. “Then what would the Quadratics think? No no, that wouldn’t do at all. You must stay, at least until I have done my duty.”

“What is your duty and why does it involve me and math? I hate math!” Joan looked a bit miffed about it all. The tetrahedron shook his head, a very large feat for a tetrahedron indeed. “You don’t even know what math is,” he said patiently.

“Sure I do. It’s a bunch of stupid numbers lined up in a row that you have to add or subtract or multiply or divide.”

The tetrahedron sighed. “My dear, what you have been learning about numbers is only scratching the surface! Tell me child, what have you learned about irrational numbers?”

“Only that they go on forever and are very troublesome to write or think about.”

“Dear dear dear. However will I accomplish this? Here, come along with me I have plenty to show you.” The tetrahedron ambled away, in a sort of tipping motion, rocking himself between his three base angles. Joan thought about not following him, but then where would she be? Lost in a big blob of color with lots and lots of shapes, that's where. She hurried to catch up with him.

“So what exactly is all this about? I don’t care at all for math. What could possibly be so interesting about it anyway. There isn’t any point to it!”

The tetrahedron stopped. “No point to math? Goodness, but math is the essence! How would you survive without basic mathematic skills? All baking would cease because how could you combine 1 egg with 3 and 1/2 cups of flour if you could not count or add? How would you know how much time you could play before you had to leave for you lessons? How would you be able to play sports if you could not keep track of the score? How could you build a house that was straight up and down? Math, little one, is everything.”

Joan grimaced. “I suppose. But why is it necessary to know about irrational numbers, or that ?rˆ2 is the area of a circle?”

“Well, it isn’t ‘necessary’. But it gives one a much larger understanding of the universe, and a great feeling of upperclassness, to know these things. Besides, some jobs require these skills.”

Before Joan could say anything to that the tetrahedron pointed to a line some distance away. “You there! Come here please, I am educating a young person in the way of Mathematics.” In no time at all the line was standing next to them. “Mark yourself please.” The tetrahedron requested benevolently. Soon Joan had a number line lying down in front of her. “Now miss . . .”

“Joan,” Joan said.

“Good. I am called Philos. Joan, you see the number line, correct? Point to the part of the line that represents one inch.”

“That’s easy,” Joan said arrogantly, and placed her finger on the 1 on the line.

“Good. Now place your finger on the point that is one foot on the line.”

Joan placed her finger on the 12. “You can’t trick me!” she said triumphantly.

“Apparently not. Now, Joan, place your finger at 2 and 1/2 inches.” Joan did so, carefully placing her finger on the mark in between 2 and 3. “Very good! Now, for the tricky part. Place your finger on the square root of two inches.”

“The square root of two inches! That isn’t possible, Mr. Philos! That number goes on forever! It can’t possibly have a place on the number line because the number doesn’t have an end. How can you know where it goes?” Joan was all flustered.

Philos smiled again. “Alright, I suppose I will have to prove to you that there is indeed a point on the line that signifies the square root of two inches. For conveniences' sake, let us expand our number line and try to place the square root of two feet. Sir line, could you?” The line expanded to double length, and marked out the feet as well as inches. “Thank you. Now, you have practiced with the area of squares, have you not?” Philos inquired.

“Of course,” replied Joan haughtily.

“So if a square had a side of one inch, what would its area be?”

“1 square inch.” Joan replied easily.

“And why is that?”

“Because to find the area of a square you must multiply the width and length together and add the unit ‘square --’ depending on what the original units were.”

“Good! So now let us suppose we have a square that is one foot wide and one foot long. You agree that the area of this square is 1 square foot, correct? Well suppose you put a diagonal line across it, and then made a square with that length. Can you picture it in your head?”

“Yes, but it is a bit tricky.”

Alright, now if you were to connect the corners of that square, and make four triangles, two of those triangles would add up to the original one foot by one foot square, correct?”

“Why?” Joan inquired, confused by all these mental pictures.

“The side of the figure was created by drawing a diagonal across our original square. So the triangle that becomes of that diagonal is one quarter of the larger square, correct? Here, let me draw it for you.”

This is what he drew.

*

“Ahh! That makes sense. So the quarter of the square that is on its point is equal to half of the square you colored red. So therefore half of the square standing on its point is equal to the whole red square.” Joan said proudly.

Philos nodded excitedly. “So you’ve found it out! So since that is true, the area of the square that is on its point must equal twice of the area of the red square. In other words it equals two square feet. So therefore its sides must, when multiplied together, equal two. The sides are also all equal, since it is a square. So what is the length of the side?”

Joan’s mouth opened in horror. “No! It isn’t possible! It can’t be!”

“But it is. The length of the side of the square is the square root of two, because the square root of two multiplied by the square root of two equals two.”

“That is not POSSIBLE!” Joan moaned.

“You can see the line for yourself! But now I have one more thing to show you. Since it can be made into a line, you might assume you could therefore measure it with something else. Tenths of a foot perhaps. Line, please mark the square root of two feet and tenths of a foot.” The line neatly displayed them. Joan saw that the mark for the square root of two was in between one and two, and almost one and four tenths, but it the mark for one and four tenths was a little bit smaller than the square root of two.

“As you can see, it isn’t measured by tenths of a foot. Four tenths go into it, but the fifth one goes past it. So you might try it by hundredths. Line, mark the hundredths.” He did so. “Ahh, so one and four tenths and one hundredth of a foot is not quite there, but two hundredths goes past it. Now try thousandths.” Again the line marked itself appropriately. However, Joan couldn’t distinguish the thousandths from eachother. Philos saw her squinted eyes, and adjusted appropriately. “Line?” Philos inquired. “Could you mark out thousandths of a mile in relation to the square root of two miles?” The line obliged, and Joan saw that four thousandths past the one hundredth marker was not quite there, but five thousandths went past it. “Those don’t line up either,” she complained.

“No they don’t. Nothing ever will, as a matter of fact. No matter how many little pieces you cut it into, none of them will ever line up exactly with the square root of two marker. There is a segment of line that equals the square root of two feet, but it cannot be broken down into smaller equal segments.”

“Stop! Enough! I won’t do it anymore!” Joan’s head hurt very much as she tried to reshape her idea of mathematics.

“That’s alright, we’re done now.” Philos tilted away, whirling and twisting madly. Joan once again felt a drowsiness overwhelm her, and fell asleep on the floor.

When she awoke she was in her room again. “Did you have a nice nap?” her mother called from below. Joan thought about it. Her head still hurt. But before she could stop herself, she found herself saying, “Yes. Yes, I did.”

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