The Geomagician's Apprentice

by L. Amaya Becvar

This story is dedicated to my friend and mentor, Ted Mackay, who introduced me to the synergistic world of R. Buckminster Fuller.

Outside the black wind howled, trying desperately to squeeze in through the cracks of a shadowy castle wall. Inside, by the light of three whitewick candles, a boy leaned over a page of handwritten work. He was scribbling furiously, a cloud of graphite dust following the trail of his busy pencil.

"There!" Ray said with a flourish. "That does it. Oh, the Geomagician will be so proud of all the work I've done!"

One of the candles sputtered and died, slipping into a pool of wax. This startled the young apprentice. He had been so caught up in his project that he hadn't realized how long into the night he'd been working. He checked the status of the other two candles on his desk and noticed that their flames were burning low, too. Ray felt a twinge of guilt. He'd been through nearly a dozen candles since the Great Geomagician had left on his mathquest.

Ray had been left in charge of the Geomagician's Lair, which was turning out to be an enormous task. Not only did he have to make certain that all of the formulas and equations were dusted and polished daily, but he was also expected to feed the Giant Abacus, and keep the Geomagician's stock of materials in good order. This included pencils, paper, and whitewick candles, as well as assorted problem solving devices such as calculators, protractors, and cans of large pitted black olives (good brainfood, as everyone knows).

Only Ray had been so preoccupied with his project that he had burned through several of the pencils and whitewick candles in the past few days. Not to mention the number of remaining cans of black olives was desperately low. And then there was the Giant Abacus, which had been letting out low growls of hunger at odd intervals during the night. Ray knew he should attend to these responsibilities, but he was determined to solve the mind-boggling problem he was working on before the Geomagician returned home.

Typically when the Geomagician was around, he wasn't allowed to do much problem solving. Nothing as interesting as this problem, at least. "Go sharpen these pencils," "Get me some graph paper," or "Another can of black olives, please," was all Ray ever heard. One day when Ray expressed his discontentment, the Geomagician addressed him in a calm, yet firm tone.

"Ray," he began, "You are here in the NRNSL (Neither Round Nor Square Lair) to learn the fine art of Geometry Magic. You may not think that you are learning now, but I assure you, with every turn of the pencil sharpener, you are absorbing another bit of knowledge. I know you think you know enough already, but I implore you to be patient. There will be a time when the lines and shapes and angles belong to you, not just to the paper upon which you have sketched them. Geometry is deceptively simple and beautifully complex. Even the tiniest snail on the garden walk can teach you vast oceans of knowledge. In fact, I'd like you to go out and find the tiniest snail in the garden and ask him to teach you about the geometry of his shell." He cleared his throat. "And grab me a fresh can of olives before you go, please."

And so Ray had spent the afternoon pouting as a light drizzle fell on him in the garden. Digging snails out from the muddy flower beds was not what he'd envisioned his life would be like as a Geomagician's Apprentice. Where was the adventure? Where was the challenge? Ray sighed. Where was the Geometry Magic?

Over a fortnight ago, the Geomagician had left on his mathquest. King Mathematicus required each of his royal council members (of the order of the Knights of the Times Table) to go on such an intellectual quest at least once every ten years. The last mathquest the Geomagician had gone on was long before Ray had ever come to the NRNSL. He had told Ray about his adventures far out on the Polygonesian Islands, where he battled looting Pi-rates in search of the coveted Gold Stars of the Mathteacher Tribe. At sea, returning home, the Geomagician had also fought with the Giant Abacus. He tamed him with a division problem that was impossible to solve. The Abacus had since come to be the sweetest and most endearing pet in the NRNSL, when he was fed addition problems on a regular basis.

So Ray wasn't sure of the Geomagician's whereabouts now that he had set off on the latest mathquest. And he certainly had no idea of what sort of adventure the Geomagician had gone on either. The Geomagician had left in a whirlwind late one evening about three fortnights ago, hurriedly tossing some items into a knapsack. He gave Ray no word of where he was headed; only outlining to him the jobs he was to do in maintenance of the Lair. Ray had followed him around the Lair, eagerly scribbling down tasks as the Geomagician rattled them off. And then, SLAM! The Geomagician swept out of the Lair into the night, closing the heavy door behind him with a dramatic thud. As the weeks dragged on, Ray grew increasingly jealous just thinking about the Geomagician's mathquest. He craved adventure like the Geomagician most certainly would be having. He was tired of the mundane tasks that he had to do to keep the Lair running, and bored with the day-to-day schedule he had devised for himself in order to complete all the required tasks. Ray had posted the following list up on the north wall of the NRNSL:

7AM-9AM Complete a page of Apprentice Geometry Made Easy
9AM-11AM Stroll around castle grounds, sunbathe near the Geoma-Tree (Ray loved to bask under its paper leaves and lay resting on its square roots)
12 NOON Mandatory stop in the castle kitchen for a fraction of fresh R-squared Pie
1PM-4PM Chores. Polish equations, dust measuring devices, feed Giant Abacus. Purchase any dwindling supplies from the "Mathematicians R Us" catalog.
5PM Supper
6PM-8PM Paperwork, practice shape drawing and model building.

Things had been running according to schedule, until one ordinary afternoon, when Ray's monotonous lifestyle took a turn of excitement.

Ray was startled from his math book by a frantic pounding on the enormous Lair door. He opened it to find the castle Page frantically waving his arms.

"Quick! Urgent! I need the Geomagician!" The Page had demanded, looking past Ray and scanning the Lair.

Ray stuttered meekly, "Uh, I'm terribly sorry sir; the Geomagician has taken leave on his mathquest. He's been away for the past three fortnights."

"Oh Matrices and Mandelbrot Sets! King Mathematicus will be most upset. Most upset indeed. We were certain the Geomagician hadn't left yet."

Ray looked around the Lair helplessly. The Giant Abacus rattled his beads. "No, he's gone sir, and I've no word of when he'll return."

The Page sighed quite loudly. "Well, leave him a message then." Ray hurried to retrieve writing materials. He was trying to appear as a responsible apprentice, but he was shaking with nervousness. He had never expected this to happen!

"Message reads as follows:" the Page began.

Dear Geomagician,

We are in grave need of better housing for the King. The castle is old and crumbling. Cold winds howl down the hallways at night, seeping in from the cracks in the walls. (The Page had a flair for the dramatic). The King has developed a severe case of bronchial polynomialitis and has been unable to sleep soundly for several weeks. His normally logical temper has grown quite irrational.

We, the Knights of the Times Table, request that you complete a design for a new castle. The design should be efficient and modern, providing the King with the utmost of comfort. In addition, please keep in mind that the King's budget is extremely tight this year, and he can only spend a limited number of counting-stones.

("Last year's crop of numbers was rather low, as you recall. A lot of them were not prime," the Page had explained to Ray).

He continued,

You have until the end of the next fortnight to submit designs for a new castle under the given specifications. Please do your best. Additionally, King Mathematicus advises you to use a bit of Geometry Magic in your design.

Signed, His Majesty, and The Knights of the Times Table, Royal Order Summa cum Laude, yada yada yada.

The Page sighed with a snooty air. "That is all."

Ray looked up from his writing. "But sir," he protested, "the Geomagician has gone on his mathquest! I have no idea how long he will be away. Won't you tell King Mathematicus that the Geomagician is not available?"

The Page looked at Ray narrowly. "Apprentice, King Mathematicus is only a fraction of his former self. He cannot be bothered with such trivialities."

"But it's not a triviality!" Ray raised his voice in desperation. "Don't you understand that if the Geomagician isn't here he can't possibly design a castle?!"

The Page rustled his corners. "Please control your outbursts. I am only here to deliver a message, not solve all of the problems of a young and obviously inexperienced apprentice. That is all. Please convey the message to the Geomagician as quickly as possible."

And with a calculated flip of his edges, the Page had left the Lair.

That night Ray couldn't sleep. He sat up late into the night; his young mind was racing with possible solutions. Should he request council with King Mathematicus? No, Ray was intimidated enough by the King when he was in good spirits. He couldn't imagine trying to explain to a sick and ill- tempered King that his new castle couldn't be designed. Should he set off and look for the Geomagician himself? Ray had begun to pack his knapsack before he realized that wouldn't work either. As much as he was tempted to go on such an adventure, he had no idea where the Geomagician had gone. For all Ray knew, the Geomagician might have even left the Kingdom of Mathalot completely.

Ray slumped in his chair, as droopy as a parabola. I don't even know how to contact the Geomagician, he thought sadly. Unless, that is, it involves some kind of Geometry Magic I haven't learned yet.

Ray felt a wave of impatience come over him. Argh! I know I've only worked in the NRNSL for a few months. But the whole time it's been nothing but boring, ordinary math! The Geomagician has taught me all of the names of shapes and how to draw them. But even a number farmer knows that the wheels of his wagon are circles! The Geomagician has taught me how to use Calculator and Ruler and Protractor. But even a cabinet-maker knows how to measure lengths and angles! When will I learn the magic of Geometry? He sighed. Probably even when the Geomagician gets back he will have me doing more pencil-work and snail-watching. No excitement.

Then Ray had a fantastic idea. What if I were to design the new castle for the King myself? When the Geomagician returns, he'll be so proud of me that surely he will know I'm ready to start learning Geometry Magic!

And so, with avid enthusiasm, Ray had set about working on the design for the new castle. Piles of crumbled paper lay scattered around his worktable. After three days and nights of calculating, measuring, and drawing, Ray's finalized sketch for the new castle lay in front of him. As he dusted the eraser shavings off from the graph paper, he admired his work. He rubbed his weary eyes as another one of the whitewick candles flickered and went out.

Turrets and walls covered the page. Ray had drawn strong stone structures that gleamed in an imaginary sunlight. True, his drawing didn't look much different than the original castle, but it was newer, sunnier, and the flags were more brightly colored.

And as Ray proudly admired his work, something strange began to happen upon the graph paper. His drawing began to move. The entire castle seemed to gather itself up and heave an enormous sigh, collapsing into a heap of dust and crumbled rocks. Ray blinked and shook his head. All that remained of his hard work was a sagging, old, drafty castle.

It's true! Ray thought glumly, I just drew the same old thing.

Defeated, Ray took off his apprentice hat and set it beside him. He ruffled his hair. He had no idea what to do. It was already so late, and he was so very tired... Ray laid his lead down on the worktable in front of him.

Suddenly, Ray felt light and airy, and he seemed to be drifting. His head was full of misty morning clouds. Slowly he looked up from his folded arms. Blinking, Ray found himself in a hazy meadow. Was it morning? He could hardly make out the blades of grass beneath his feet. Ray walked forward through the fog, hoping to find something recognizable. A large, imposing shape appeared before him.

As Ray approached the shape, he realized that it was the oaken worktable upon which he had been drawing his castle sketches. Its surface was empty, and it was oddly different than Ray's worktable at home. The desktop appeared to swirl like the fog around him. There were no pencils on the worktable, so Ray opened the drawers one by one, searching for materials. In the bottom drawer he found a heavy cherry wood box.

Ray took it out and ran his fingers over the surface of the box. It was cool and smooth. Slowly he lifted the lid and found a long, slender object lying in the soft blue velvet lining the box. It was a pencil made of the same dark cherry wood as the box. Ray marveled at the delicate carvings along sides of the pencil. Many symbols appeared; some familiar and some strange. All relating to math, he guessed.

A thought occurred to him: Could this be the Geomagician's magic wand? It was heavy and imposing in his hand when he lifted it from the box. Ray touched the tip of the wand to the swirling worktable surface. The surface rippled like water where the tip was set down. A tiny dot remained when he lifted the wand.

It's a point, Ray thought. It had been his first lesson with the Geomagician. A point, the simplest geometric concept. An infinitely small location; a shape with no dimension.

Ray touched the Geomagician's wand to the surface and created another point. A flash of light connected the two points. Now I've made a line segment, he mused.

Words began to rise up from the depths of the worktable, coming slowly into focus. "ONE DIMENSION," it read.

Yes, a line segment has one dimension, Ray thought to himself. Length. He was starting to enjoy this game; he was good at it at least.

He touched the wand a third time. Another point appeared, and two line segments sprung from the endpoints of the first line segment, connecting to the point he had just drawn.

Easy, it's a triangle, he thought with a smile.

"TWO DIMENSIONS," appeared in cloudy text on the surface of the worktable, beneath the triangle.

This figure has length and width, Ray thought. He touched the wand to form a fourth point. Next comes a square, he predicted. This game was getting to be too simple. Ray had learned how to classify shapes in his first month with the Geomagician.

But instead of a square, three more line segments grew out from the endpoints of the triangle, all connecting to the fourth point. The whole structure began to rise and hover above the surface of the desktop. Three more triangles had been formed in addition to the first.

"THREE DIMENSIONS" floated up from the depths of the worktable. A delicate breeze blew and set the shape gently rotating.

The structure waited patiently, almost seeming to ask Ray for more. But he was confused; he didn't know what his next move should be. Just guessing, he touched the end of the wand to the object and murmured, "Four dimensions."

It responded with a flash of light. The object, once hazy, burst out of the desktop and sat plainly in view. The sides appeared to fill in and become solid. The worktable's surface stopped swirling and looked like ordinary wood again. The shape sat upright on the surface.

Of course, Ray thought. Here in the Kingdom of Mathalot we exist in the realm of Four Dimensions. I have introduced this object into my reality!

Ray reached out to touch it. It looked real enough, like a chunk of shaped metal. A silvery glow from inside the object stopped his extended hand.

"Good work, young apprentice," Ray heard spoken near him. The object flickered with each syllable. The voice seemed oddly familiar. He turned and glanced around the clearing, looking for another mathema-citizen who might have spoken to him. The surrounding mist had lifted, exposing forms of trees around the perimeter. Yet he could see no one else in the clearing.

"Thank you," Ray finally replied, glancing around him, unsure of where to direct his voice.

"It is I who should thank you, good friend," the voice responded graciously. "You have manifested me into the Dimension of Four and for that I am deeply grateful."

Ray now realized who was speaking: the object in front of him! It seemed peculiar to be speaking to a metal shape, so Ray asked, "Who, or what, are you?"

"I have a different name in different dimensions. With no dimensions, I am Point. Given one dimension, I become Neverending Line. Unless, of course, you restrict me to exist between two points, and then I become Line Segment. In two dimensions I am the Infinite Plane from which all polygons can be cut. Unless, of course, you restrict me to exist between three points, in which case I become the simplest enclosed polygon, Triangle. If you give me another dimension, another point, I become the simplest polyhedron, Tetrahedron. Four sides, all triangles."

He went on. "And you gave me another dimension, and now I am part of your world."

Ray cocked his head, his mind spinning. "But what did I give you which brought you to life? I didn't set another point down--" He knitted his brow, trying to recall the steps he had taken.

Tetrahedron paused dramatically. "Apprentice, you gave me the dimension of TIME."

"Gosh," Ray replied. "Time is a dimension? Are there other dimensions beyond that?"

"Yes, of course! Many more. Yet you cannot comprehend them, being human, being 'of four dimensions.'"

"Oh," Ray said, and then had a terrible thought. Time! He only had a limited amount of it to complete the design for the castle. He needed to get back to the NRNSL!

"Tetrahedron, can you tell me why I'm here? I understand about four dimensions now. But you see, I really must get back to the Neither Round Nor Square Lair. I don't really have time to be here. You see, I'm having a lot of difficulty with this really important problem..."

"Yes, I know. The King's castle." Tetrahedron glowed warmly, as if to soothe the anxious apprentice.

Ray looked at Tetrahedron questioningly. "You know about the King's castle? Well, I suppose that doesn't surprise me. You seem to know everything." He paused, considering. "Hey, perhaps you can help me find the Geomagician!"

Tetrahedron responded with feigned shock. "Now why ever would you desire that?"

Ray sighed. "I need his help. My castle design is a flop. Literally."

"But Ray, what good would it do if the Geomagician were here?"

"Easy," Ray said, "he could design King Mathematicus' castle!"

The Tetrahedron paused. "Ray, do you think the Geomagician would be teaching you anything if he just solved all of your problems for you?"

"Yes...well, no. I suppose not." Ray sighed. "But I still don't know what to do."

"Okay, then I'll give you a little bit of guidance." Tetrahedron offered. He began to glow with a warmer intensity. "Ray, I would like you to manifest a cube."

"A cube?" he asked, "Made the same way I made you? All right," Ray agreed, and set about his work.

First Ray made a point, four line segments, a square, the five more squares. Then in a flash, FOUR DIMENSIONS, and it became real. Yet the Cube remained a skeleton of line segments, never filling in like Tetrahedron.

"Hello?" Ray leaned over and asked the Cube. Nothing.

"Go ahead, apprentice, touch it," Tetrahedron encouraged.

As Ray reached out to pick up the Cube, the whole structure sagged and collapsed to one side.

"Oh, no!" Ray cried out. "What did I do wrong?"

"Nothing, Ray." Tetrahedron chuckled. "A cube isn't very strong. Watch what happens to my skeleton."

Tetrahedron turned himself into a structure of connected lines. When Ray picked it up, it remained rigid, even when he pulled and pushed on its angles and edges.

"A triangle is strong," Tetrahedron explained, "able to resist force by spreading it out over the lengths of its edges. A square, however, focuses all of the pressure on its joints. It is not a very strong structure at all."

Ray nodded.

"Likewise, a tetrahedron (so simple, so perfect!) is rigid. A cube, twice as complex, isn't a very stable structure."

"So...why should I build a castle out of cubes?!" Ray added excitedly.

Tetrahedron glowed. "You've got it. A cubic castle has to be made with heavy, clunky rocks and lots of material to increase its strength. A tetrahedral castle would be simpler, lighter, more efficient to build and easier to repair..."

"...And that's just what the Knights of the Times Table want!" exclaimed Ray delightedly. "So, my basic building block is the tetrahedron. Great. I'm ready to go home now, Tetrahedron!"

"All right, Ray. Now you're beginning to know the importance of understanding the rules and limitations of your world. Take this knowledge and make it your own."

Ray nodded in assent. His brain felt terribly full, so he turned and said farewell to Tetrahedron and the worktable in the clearing. As he began to walk to the edge of the forest, fog swirled in from between the fingers of slender trees. The fog engulfed him, and he grew terribly drowsy..."

The boy lifted his head from the worktable. He squinted as his eyes met a sliver of morning sun that streamed in from the skylight above him. His red hair stood up in an amber corona all around his head. I'm back in the NRNSL, he realized, noticing his surroundings. Or did I ever leave?

The three whitewick candles had all burned away, leaving long fingers of wax spread over the empty pages in front of him. Has this all been a dream? he wondered, shaking the drowsiness out of his head.

But one definite clue made Ray certain it had not been a dream. As he uncurled the sleepy fingers of his left hand, he noticed that he held a small, intricately carved wand. It was smaller than the wand he had used to create Tetrahedron, but lovely nonetheless. What was most striking about the wand was that attached to its end was a tiny crystal tetrahedron. And when Ray touched the end to his pages of graph paper, it left a thin line much darker than a pencil would. Now I have my own wand! he thought delightedly.

Ray set about his work for he knew he had much to do. And his wand seemed to know just where to go, and what marks to make. The sun raced across the sky as he worked that day. Ray stopped to break only for olives and an occasional glass of lemon-flavored seltzer water, which had soon become his favorite drink.

He found himself near the completion of his new design at the day's end. Ray smiled at the improved castle which stood before him, flattened into two dimensions on paper, but gloriously four- dimensional in his mind.

Tetrahedrons were everywhere in the design; in fact, every room in the great castle was complemented by the strength of the triangle. Ray was especially proud because he had been able to design a structure with a great deal less material than his original designs. According to Ray's calculations on the Giant Abacus, his new designs would be saving the King almost 300,000,000 counting-stones.

But in the flickering glow of a whitewick candle, something seemed to move on his paper yet again. Ray crouched closer, studying the surface. He squinted his eyes in the dim evening light. A tiny man moved from room to room of the design. Wearing a large angular crown, the man looked rather like King Mathematicus.

"Ouch!" the little man exclaimed in a tinny voice. Ray jumped at the high-pitched noise. The tiny king appeared to have hit his head upon one of the slanted walls. His itsy-bitsy crown toppled to the ground with a mouse-sized clatter.

"Blitherstat! There is no room for anything in here!" the teensy king said in annoyance. "Where am I going to put the Pythagorean Throne? Blast these angled walls!"

Watching in horror, Ray smacked his forehead with his hand.

Where did I go wrong? Ray wondered in frustration. The castle will stand up, and will cost much less to build...but there is no room to live inside of it! The apprentice looked angrily at the wand in his hand, blaming it. But then Ray realized that he himself was the one who told the wand which marks to make. If there was a problem with the design, it was entirely his fault.

In one furious swoop, Ray gathered up his castle sketches and crumbled them up. The Giant Abacus growled questioningly in the corner of the Lair. "Oh hush!" Ray shouted impatiently.

What am I going to do now? he thought, shaking his head. His frustrated anger turned to discouragement. He was too exhausted to be angry anymore. Ray's head grew heavy with defeat and he rested it upon his folded arms.

I'm so tired of failing, he thought. Maybe I should just give up and wait for the Geomagician to get back. I'm just no good at this. I must be too young, or just plain dumb...I'm never going to be any good at Geometry Magic.

Tears of self-pity began to well up in the boy's deep brown eyes. They fell onto the empty worktable with a patter as soft as raindrops. Quietly, Ray began to cry, thinking of his loneliness as an apprentice. He thought of his parents, who had toiled in the Decimal Point mines to support him as a young boy. What if they had asked him to design them a home? Would his fail them, too?

Soon, he began to feel his body being lifted into a dream again. He continued to cry, not wanting to stop thinking of his plight, and of his family. When he finally paused, he realized that his sobs had been drowning out the sound of ocean waves. He lifted his head to take in his surroundings.

He stood on a great cliff overlooking a wide sapphire sea. Unfolding above him was a deep velvet sky sprinkled with glass-shard stars. In front of him lay the magic worktable.

"You have cried yourself an ocean, Ray," a female voice said as gently as the murmuring waves.

Ray turned his eyes to the glittering heavens, looking for the source of the voice. His face was bathed in the alabaster glow of a wide-faced moon. Then he heard the voice again.

"Ray," it began, "you know how to draw shapes and make them to come to life."

"Yes," he agreed.

The voice questioned him. "What is the polygon with the fewest sides?"

"Why, a triangle, of course."

"And a tetrahedron...?" the voice asked.

"...is the three dimensional shape with the fewest sides. Of course it is. But my castle design..." he said, disheartened.

"I know. It didn't work well," the voice soothed him. "But my dear Ray, that is simply because you have forgotten to consider volume."

"Volume?" he asked, confused.

"Yes. I'll show you," the voice responded. "Draw both a cube and a tetrahedron made up of line segments of the same size."

"Okay." Ray took out his small magic wand and created each shape on the worktable.

"Now fill each with water."

Ray reached down and scooped up a bit dark ocean water into the cube, and then into the tetrahedron.

The voice asked, "Which holds more of the sea, Ray?"

"Why, the cube does," he commented, surprised. "But I thought tetrahedrons were the best shape!"

"The most stable shape, Ray." However, tetrahedrons hold less volume than cubes. That is the secret to their strength and efficiency."

The boy shook his head, not understanding. "So I should have made my castle with cubes? And tetrahedrons together? I'm confused!"

"Don't be." The voice consoled. "Ray, what is a two dimensional shape with five sides?"

"Um...Pentagon," he said proudly.

"And the shape with six sides?"

"Hexagon. This is easy!" he remarked.

"Okay, good, Ray. Now, I want you to draw me a shape that has an infinite number of sides. So many that you can't count them."

Hesitantly, Ray held the magic wand above the worktable. He looked up at the moon questioningly. Then, something made sense as he regarded the round white orb above him.

"Aha! It's a circle! Circles have so many sides they're too small to count!"

"Good, good!" the voice praised. "And now in three dimensions?"

"Sphere!"

And with that the pearly moon began to descend upon him, floating down like a soap bubble. But as it drew closer, Ray realized that it wasn't the moon at all, but a flawless crystal ball. It glowed with a soft light from deep within.

The crystal Sphere began to speak. "Yes. Spheres have the greatest volume for a given size. In a sphere there are no edges or corners, no wasted space." Ray thought for a moment. "So, I need to combine the strength of the tetrahedron with the volume benefits of a sphere!"

"You've got it, Ray! Now just close your eyes, and believe in yourself!"

He did. Once again he found himself back in the NRNSL to meet the new dawn. Enthusiastically, he tore off a clean sheet of graph paper and touched the end of the small magic wand to his tongue in thoughtful preparation. With a smile, he noticed that a tiny crystal sphere now sat above the tetrahedron on the tip of the wand. Eager to use his new knowledge, Ray drew a large sphere with covered his paper.

Spheres, spheres everywhere, he thought gleefully. His tiny wand came to an abrupt halt. Wait...where would a door go? What could keep the sphere from rolling away? Oh, no! With multiplying frustration, Ray tore the paper from the worktable. He ripped it in half and threw the pieces to the floor. Holding his chin in his palm, he examined the torn halves on the cold stone floor.

That's when he understood.

Ray finished his design approximately five hours and two olive cans later. Just as he popped the last olive into his mouth and added the final lines to the drawing, he heard someone fiddling with the heavy Lair door.

With delightful nonchalance, in strode none other than The Great Geomagician himself. His purple Geomagician's cloak was torn and dirty, and his overgrown beard stretched nearly to the floor. But a hyperbolic smile stretched across his face.

Ray stood bolt upright, almost as if he had been faced with the Shepp-Logan Phantom.

"You're back!" he exclaimed. "Thank Euclid! I need your help desperately!" Ray blurted out.

The Geomagician looked at Ray questioningly. "Back? I never went anywhere," he said. "And it looks like you're doing just fine on your own, apprentice," he added, striding over to examine the pages on the worktable more closely.

But Ray wasn't appeased. "I could have used your help these past few weeks. King Mathematicus' snooty Page came in and demanded that I give you a message to design a new castle for King Mathematicus. He wouldn't even listen when I told him you were away."

The Geomagician looked puzzled. "But I wasn't away."

"Yes, you were! Your mathquest, don't you remember?" Ray couldn't understand the Geomagician. Could he so soon forget?

"Oh, Ray," he said with a chuckle. "You thought I was away. But I really never left at all."

Ray was shocked. "You were here all along? But I needed your help!" He paused. "And you told me you were leaving on a mathquest. I thought I had to design the castle myself because you weren't here to do it..." he said, then looked down, realizing that he sounded quite presumptuous.

"I wouldn't neglect my duties to the King, Ray."

"Oh, of course. I'm sorry. I guess I shouldn't have assumed..." He felt terribly foolish. "You mean you already have a design?" he asked quietly, defeated.

"No," the Geomagician replied. "You do."

Ray was confused.

The Geomagician chuckled. "You've been learning, Apprentice, haven't you?" he asked. He pointed to Ray's sketches, and pulled up a wooden stool. "Show me what you've done here."

"Well," Ray began, with growing pride, "by combining the strength of the tetrahedron with the volume benefits of a sphere, or half sphere, actually...I propose to build a dome made of tetrahedral struts."

With a smile, the Geomagician asked, "Did you enjoy your dreams, Ray?"

"My dreams? How did you know about...hey!" Suddenly it made sense to him. "Of course! You were the voice of the Tetrahedron!"

"Yup." The Geomagician grinned.

"Who was the voice of the Sphere?"

"Who do you think, Ray?" The Geomagician got up and strode over to the Giant Abacus, giving her beads and affectionate ruffle. "Our good friend Abby, of course!"

The Giant Abacus purred with contentment. Ray felt a twinge of guilt. He'd been neglecting the Giant Abacus much more than he should have.

"Abby?" Ray inquired. "Gosh, I never knew she could talk!"

"Abby uses thought-speak, Ray. You just never listened to her before."

"Hello, Ray," the boy heard spoken in a sweet voice in his head.

"Oh, wow," Ray commented. "I felt so lonely, and you were both here, all along." He paused. "Wait, what do you really mean, 'You never went anywhere?'" Ray asked the Geomagician.

"I've been in and amongst other dimensions, of course. I've been around, offering guidance here and there, in my multi-dimensional way."

The boy looked puzzlingly at his mentor. "But why couldn't you have just helped me from this dimension?"

"Well, in order to truly learn something," the Geomagician began, "you have to develop the will and determination to learn entirely on your own. You decided on your own to design the castle. It was a challenge you made to yourself. And," he continued wisely, "in order to make knowledge entirely your own, you have to use that knowledge to bring something, like an idea, into reality."

It was true. Ray did feel like he owned his understanding of lines and shapes and volume and...

"Oh!" Ray asked, "So what do you think of my castle design?"

The Geomagician winked. "It's terrific! I couldn't have designed it better myself."

Ray submitted his design to the Knights of the Times Table shortly thereafter. After convening over crumpets and N Factorial Tea, the KTT's unanimously approved the blueprints and called for the castle's immediate construction. As Sir Calculus put it, they "adored its simplicity and unconventional mathematical genius."

Thus, with much fanfare, the new castle was completed and King Mathematicus moved in. His sleep patterns greatly improved and his rational temperament returned, much to the relief of the mathema-citizens of Mathalot. In fact, Ray's castle design was so popular that soon tiny domes of similar design began to pop up all over the countryside. Ray was promoted from Apprentice, Second Class, to Royal Architect. He still continued to work with the Geomagician (and Abby), though, who made it a top priority to ensure that Ray learned the fine and mysterious art of Geometry Magic.

Did you?

Author's note: This story was inspired by the ideas of R. Buckminster Fuller, architect, engineer, poet, and visionary thinker. "Bucky" invented the Geodesic Dome, which is the inspiration for Ray's castle.