She frowned and stared at the wall behind Tahirah. Another vertical reflection symmetry—that would be the third. But no, it wasn’t right. One tile in the band was twisted, so it wasn’t a perfect reflection. As she watched, the tile wiggled and turned before her startled eyes. Now it matched the rest. What was that? She stared again, not sure that she had truly seen it move.
A noise came from Layla’s embroidery basket, then right after, another noise far away, almost like a roar. The basket bounced. Layla dropped it and backed away. Ara grabbed it, hugging it to herself. Tahirah looked at it in surprise.
“Arrrrrrr…Arrrrrra...” came from the basket.
Its lid thumped up and down. They all stared. “What have you done, child?” Tahirah asked Ara.
“I...I can’t tell you,” she stammered. The racket from the basket was getting louder.
“As Allah is good, I think you truly must tell me. Daughter of the harem, something is very wrong here,” exclaimed Tahirah. Something is very wrong indeed, she thought as she watched the basket’s lid open and a green snake, twice as long as Layla’s arm and half as thick as her wrist, slithered out. Layla backed up against the wall and covered her eyes with her hands.
“Oh, no,” cried Ara. “Where’s Suleiman? Please, as Allah is merciful, let him not have been eaten by a snake.” Ara riffled madly through the basket. Embroidery scattered everywhere.
The snake raised its head and spoke. “Arrrrra, I promisssssed.”
“A transformation,” whispered Tahirah in a shocked voice and watched the snake wind its way across the floor. “Is that Suleiman?"
“No, Suleiman is a lizard,” Ara wailed. “Not a snake. He must have been eaten.”
“I think not,” Tahirah said carefully. “Truly, that is your servant before us in snake form. And he calls you. Tell me, what did Suleiman promise?”
“Nothing,” Ara said, sniffling. Then she remembered. “Well, just a silly thing about mathematics. But what happened to the lizard? Why did he change?” She saw her cousin shrinking into the wall. “Layla’s afraid of snakes.”
Layla, pressed up against the wall, peered out from between her fingers.
Reaching out her hand, Tahirah said, “I think we are all going to have to be very brave here, Layla. Come and sit next to me. Suleiman has been bespelled, and this will take all our efforts to resolve.”
“I can’t.” Layla leaned farther away.
“Yes, you can. I’ll help you be brave. Don’t think of him as a snake, but as your friend, Suleiman, who’s in trouble and needs your help.” Tahirah’s voice enveloped the girl like a protective hijab.
Layla, still eyeing the snake, slowly scooted forward to sit at Tahirah’s side, and Tahirah turned to Ara. “Now, I want you to tell me everything you know about how this happened. I particularly want to know about this promise.”
Hesitantly at first, Ara told the story of the wazir. She told about the frogs, the broken mirrors, the tiles changing, and finally about Suleiman turning into a lizard. Tahirah listened gravely with Layla tucked into her arm. The snake had wrapped itself around Ara’s wrist like a bracelet and lay quietly there.
“I see. This explains much. Did Suleiman speak to you before, in his lizard shape?”
The girls looked at one another. “No, he just lay curled in a ball,” Ara responded finally.
“And the promise?” Tahirah asked. “How did that come to be?”
“He was teaching me band symmetry, and I made him promise to teach me more. I remember he swore on his tribe’s honor,” she said. “Would that mean something?”
Tahirah sat and thought. She looked at the snake and then at Ara. “What exactly was the promise?”
“He promised to teach me all seven band symmetries. He had just taught me vertical reflection symmetry and sent me to find three examples before he would teach me the others. But he promised he would once I found three examples. And they had to be correct. Just now I saw that last one over there, but it wasn’t right, it was twisted,” she said pointing, “and then it changed and fixed itself.” She looked to see if Tahirah believed her, but the mathemagician’s face gave nothing away.
“And then I asked him about the tile that I saw change and he said it wasn’t possible. That if a tile could change, so could he—Oh, did that have something to do with him becoming a snake? The symmetry fixing itself?”
They all turned to look at the symmetry. It was a reflection. The gold shapes flipped over and over across the wall.
“See, each one is the mirror of the one before and they flip over a vertical line,” Ara said. “After it changed, the thumping began.”
Tahirah pointed to the tile, “It repaired itself, or did you repair it? The Alhambra must listen to you indeed.” Tahirah was quiet as she stared at the wall. The tile was now smooth and flat.
“How long ago did Suleiman make you this promise?” the Sufi asked.
“It was about two days after you arrived. Suleiman was angry with me for being outside the palace to watch the parade.” Ara stopped, aghast at her unplanned confession.
“Yes, I felt you that day.” Tahirah closed her eyes. “Your aura radiated excitement, fear and joy. Such a strong presence you are, child. I remember being curious about a girl so daring and bold. Much like me when I was your age,” she said with a slow smile. “Now, let’s see, let me check my journal for the exact date.” Pausing, she rose and walked over to a little niche in her room. She picked up a small book and quickly flipped through its pages. “Yes, as I thought. I arrived three weeks ago today, so Suleiman was bound by his promise exactly twenty-one days ago. How interesting,” she murmured, calculating numbers in her head. “There are seven band symmetries, and Suleiman asked you to find three examples of the first symmetry.” She glanced at the girls. The snake wrapped about Ara’s arm and turned in agitated circles.
“Promissssed, Arrrrra. You helpppp,” Suleiman hissed, looking at Tahirah.
“Hmm. We have a two-fold problem,” she said finally. “Both the Alhambra and Suleiman are affected. We all saw what just happened. You found two examples of vertical symmetry, but this third one was broken, tainted by the magic of the wazir. When you saw it and noted its wrongness, it was able to heal itself. Then Suleiman also changed from a lizard into a snake.
“This is encouraging. The bands of magic holding him are more fragile than the wazir thinks. It is also hopeful that Suleiman now speaks. He remembers being human and fights the spells.” She looked off and gathered her thoughts.
“We need to continue your education in symmetries,” she said, nodding her head emphatically. “Because of his promise to teach you each of the band symmetries, I believe we may have a way to restore Suleiman. The wazir’s magic was incompletely drawn. A promise, especially one coupled with tribal honor, is binding.” She stopped suddenly and considered.
“The wazir tied his evil magic to Suleiman. That may yet be his undoing.” She looked intensely at Ara. “As the palace heals, so does Suleiman. You are the catalyst: the daughter of the Alhambra born of the line of the Nazrids. The Alhambra was betrayed by one of its own, the wazir, and it is distrustful.
“You must continue finding the symmetries. You need to seek out the damaged tiles for each symmetry. This reminds the Alhambra of its strength. It fights the spell, just as Suleiman does. If we are particularly fortunate, releasing Suleiman from this binding should also undo the evil the wazir has twisted into the Alhambra. Though whether Suleiman or the Alhambra will fully heal, I cannot say.”
Ara sat unmoving, struck by the enormity of the task placed before her, and laid her free hand on Suleiman.
Tahirah was silent for some time before speaking. “Well, we must do what we can. Magic, as you know, is as logical and ordered as mathematics. To reverse this spell, time is halved and then halved again.”
She looked up at the two silent children, their eyes wide in worry—and yet a touch of hope flickered there. “The Alhambra’s magic is at risk. If this evil
continues, the Alhambra will fall.”
Layla turned even whiter. Ara looked down to hide her fear and, seeing the snake on her arm, nodded, remembering.
“Father has been really worried this spring, but he said that the Alhambra protects its own. Is the symmetry magic how it’s done?”
Tahirah smiled encouragingly at Ara. “It’s more complicated than that, but yes, magic is buried in the symmetries, and Suleiman is now tied to that magic.
“And now, we are also working against time. If it took you twenty-one days from the time of Suleiman’s promise till you found the symmetries, you have only half that time to find the next and half again for the one after.” Tahirah spoke with quiet certainty. “We are in a race to save both Suleiman and the Alhambra. If we can pull Suleiman out of this enchantment, some of the wazir’s evil may be broken and the Alhambra’s strength regained.”
“I see,” said Layla, perking up. “Because Ara completed the first task set by Suleiman in twenty-one days, then the next task needs to be found in ten and one-half days, then five and one-quarter days.” She looked worried. “I guess you have to figure hours after that.” She wrinkled her nose.
“Yes, sixty-three hours, thirty-one and one-half hours, lastly just under sixteen hours. Each time it halves. That will mean the last symmetry must be found in under eight hours.” Tahirah looked up gravely. “We are lucky it took you so long to find all the examples of the first symmetry. A shorter time would be almost impossible for us.” She stopped for a moment, lost in thought as she gazed out her latticed window.
The responsibility dizzied Ara. “But it’s still not enough time. I could never find them all. I don’t even know the other symmetry patterns. How can I find broken ones if I don’t even know what the correct ones look like?”
“Well, as it happens, I know those symmetries,” Tahirah said with a quiet smile. “If Suleiman will wrap himself around my wrist, I can act as his…assistant. We can do this.” She looked at Layla. “And you will help as well, won’t you?”
“I can’t. I’m not good at geometry like Ara, and I’m not brave and…”
Tahirah squeezed Layla’s shoulders. “You are. Everyone can do mathematics. It may be easier for some, but everyone can master the principles. We need you. Suleiman is counting on us. Both mathematics and bravery can be learned. You must decide if it’s important to you. Suleiman needs all of us now.”
Layla ducked her head. “But it’s so very hard for me.”
“It’s going to be hard,” Tahirah said. “Hard for Ara to keep her counsel about Suleiman and not to accidentally betray herself before the wazir. Hard for me to admit I need the help of two girls to solve a mathemagical problem—and very hard for Suleiman to be a snake. He has placed his faith in us, and we hold the key to his release. We can lean on each other for strength. I’ll help you learn symmetry. Ara and you will help me learn about teaching girls. Oath?” she held out her hand.
“Oath,” both girls replied, their voices solemn, and grasped her hand in a three-way clasp.
Chapter 14
“But why did you bring us here in the first place?” Ara asked, turning her head to the side. “We thought we were being punished.”
Astonishment and hurt flashed in Tahirah’s eyes. “Do you think me an evil djinn? I hoped you would prefer me to the diet of bread and water that others suggested.”
Ara mouthed the word, “Oh.”
“I thought we could explore the Alhambra together. I would teach you mathematics and Sufi mysticism.”
Tahirah tilted her head up at the ceiling’s honeycombed recesses and winced. “The Alhambra is in pain—I can feel it—I had a vision showing a key. I believe both of you are part of that key.”
Ara reached one hand out, touching the ceramic tiled wall as if to comfort the Alhambra.
“My visions are sometimes cloudy, but this one was clear. I spoke with Suleiman not long before he was transformed. He offered to go into town for me to retrieve a scroll. He spoke with such pride about the two of you.” A sudden memory made her shake her head. “I do think that you should begin by removing the red stains from the lions’ chests. The color makes them restless, hungering for a kill. They are lions, after all. Your father wished them cleaned, and I agreed to see that it was done,” she added. “I hope you don’t mind.”
“The stone lions? They have feelings?” Layla said.
“Of course they do, though they are lions, and their thoughts are not like yours and mine. They were created long ago as defenders of the Palace of Lions, placed in its central court facing out in twelve directions, so that none could approach without their knowledge. Each lion was imbued with a trait to support the sultan.”
“Do you talk to them?” asked Ara.
“I have tried,” she replied directly to Ara. “But they have not responded. Nor is your father’s lion at his side. I worry…perhaps it’s just that they don’t trust easily or perhaps many things. We can’t know.”
“My father has his own lion?”
Tahirah looked thoughtful. “Yes, it is said each lion trails the ruler of the Alhambra, though not every ruler of the Alhambra is granted the ability to see this wonder.”
“But if you can see them, why won’t they talk to you?” Ara asked.
“They are the guardians of the castle and are wary of outside magic. I’m still not sure what the wazir is trying to do. Why is he harming the Alhambra?” Tahirah mused, then shook her slender shoulders. “Is he embroiled with the Castilians or with the Saracens or the Aragons—or some other group entirely—or is he on his own?” She closed her eyes. “All in good time, as Allah wills. We have more pressing business. I need to teach you the next symmetry. Here, Suleiman, come and wrap around my wrist so we can do this together,” she said directly to the snake, extending her arm.
Suleiman uncoiled himself from Ara’s wrist and slithered across the floor to Tahirah. “Let’s see, Suleiman taught you mirror symmetry across a vertical line, right?” The snake rested unblinking in his new location. “My thought is that the second one learned should be mirror symmetry over a horizontal line. We should probably review both.” She pulled out a quill and paper and launched into her lecture.
“You remember that symmetry is a relationship of characteristic correspondence, equivalence, or identity among constituents of a system. In mathematics it is a property that generates repeated patterns. Band symmetry, which we are focusing on here, runs in one direction like a frieze or border. Euclid stated in his treatise that there is a connection between—”
“Actually no, I don’t remember what you’re saying at all,” Ara interrupted, startled by the complicated words. Layla sat beside her, looking despondent.
“No?” Tahirah asked, surprised. “That’s the way my instructor presented it to me. How did Suleiman explain it to you?”
“Well, a lot simpler. He did talk about repeated patterns though,” Ara answered uncomfortably.
Suleiman raised his head, “Ssssymtry is sssimple.”
“I see. Let’s start over, shall we?” Tahirah took a deep breath and tried to recall herself as a young girl. “I’ve got it. Symmetry is about two things, pattern and motion—in particular, which motions a pattern can make and still remain the same.
“Let’s draw some simple patterns. Ara, you draw one, and Layla, you can draw the next. I’m going to explain reflection or mirror symmetry across a horizontal line.”
“Suleiman called it a flip,” Ara interjected.
“He did, did he?” Tahirah frowned at the snake in her lap. “All right then, reflection, mirror or flip—all three are names of this motion. Each of you draw something simple on the parchment. Just one thing each!”
She smiled at them. “Lovely. Those are excellent. Now we shall pretend there is a line below them that they reflect…oh, sorry, flip over. Watch. I will draw the horizontal mirror image, or flip, of each of your images. The horizon is an imaginary line that stretches left to right across a landscape,�
�� she added by way of explanation. “See how these images are now ‘flipped’ across the imaginary line below them. A perfect match, if we were to see it in a mirror.” Tahirah hesitated as she looked at each girl. Ara’s eyes were bright with comprehension as she focused on the drawing. The dawning of understanding crept more slowly across Layla’s face.
“But what about the line? How can you find an imaginary line?” the girl asked.
“Good question, Layla,” Tahirah said, then considered. “Well, you are looking for a place where the pattern would be repeated. In the drawings you just did, we placed the line beneath the first pattern. That was the easy way, starting with an object, drawing a line under it or beside it and completing the symmetry yourself. But how do you find the symmetry in an already completed object?” She looked at both girls. “Any guesses?”
“Maybe you could cover a pattern until only half of it shows?” asked Ara.
“Yes,” Tahirah agreed. “And there are other ways. Think about mirrors.”
“You could hold a mirror up to it and see if the other half of the pattern is repeated,” Layla offered.
“Good. Now, let’s continue. The vertical flip goes left or right, correct?” Tahirah ran her hand across the contented snake. “That’s what you learned from Suleiman and the kind of symmetry you were searching for. In the horizontal flip, the pretend line goes across, and the pattern flips up or down. Let’s practice the symmetry in repeated patterns. Here, I will draw a few and you tell me if they are horizontally symmetric or not.” She reached for her quill and ink. “This is a test,” she said, smiling. “Here I will present a whole row of the symmetry as you would see it in the tiling of the Alhambra.”
“That’s easy,” said Layla. “It’s a horizontal symmetry.”
“Excellent. And this one,” Tahirah said, sketching madly.
“There,” Ara exclaimed, reaching over to trace an imaginary line with her finger. “The middle of the budlike thing. If you divide it in half, the whole pattern is flipped over.”
The Stone Lions Page 6
