“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.”
“Good. And here is yet another.”
The girls looked at one another. Ara stated, “Well, I see a line that splits it in half exactly, but if you flip it over, it doesn’t match up. The triangles on the top would land in the blank space.”
“Perfect.” Tahirah smiled. “This is not horizontal mirror symmetry, just as you said. It does have symmetry, but not the ones we are speaking of today. How about this?”
“Oh, I see it,” Layla said, pleased. “That’s what Ara showed me. It’s a vertical flip. I remember she said that the line runs from the earth all the way to Allah. I see that there would be a line between the heads of the chickens going up and down.” The snake had lifted his head and seemed pleased.
Tahirah reached over and squeezed Layla’s shoulder. “I am very impressed. That was excellent. There is one further requirement for the images you search for. It must have a horizontal symmetry, but it cannot also have a vertical symmetry.”
“What?” Ara looked up. “How could it have both?”
“It is possible for both horizontal and vertical symmetry to exist in one design. But for our purposes you must eliminate any designs that you find with both, as they will not break the spell. The only symmetry it can have is a horizontal reflection. Can you remember this?”
“Yes,” the girls agreed, though Tahirah could see they both were somewhat surprised by this new requirement.
She glanced outside and tried not to show how worried she felt. “We have only one and one-half weeks for the two of you to find the horizontal symmetry damaged by the wazir’s magic. But first, the stone lions must be washed clean of the red dye. Of most immediate importance, we need to protect Suleiman in his snake form. I will do some mathematical searching to find out why the wazir is destroying the magic in the palace.” She stopped, suddenly aware of the lengthening shadows. “It is getting late. We must end this lesson. Such a short time to resolve a difficult spell.” Tahirah looked down at the snake.
“How do I know where the wazir broke it? It could be anywhere in the Palace of the Lions or the Palace of the Myrtles, even in the Palace of the Partal…” asked Ara. The importance of her task overwhelmed her.
“We can’t know. He is putting pressure somewhere, but the magic ripples throughout the Alhambra, finding a weak spot. There it breaks a symmetry. It could appear anywhere.
“You must not attract the attention of the wazir. If too many people know that you search for symmetries, broken symmetries, he also will know. But Layla can help, can’t you?”
“I think so.” She looked at the green snake wrapped around Tahirah’s wrist and shuddered. “Will I have to touch the slimy snake?”
“Yes, in fact, you need to hold him now. Layla, your life has been sheltered in the palace, but one who loves and cares for you is in trouble. His life is in your hands. Remember, this is Suleiman, and he’s having a very difficult time.” Gently, she placed Suleiman in Layla’s lap.
The snake curled into a relaxed S-curve. He picked his head up and stared at Layla. “Nottt ssslimy. Llayylla ccowarddd!” he hissed, bobbing his head up and down.
“I am not,” she exclaimed. “Well, maybe I am timid, a bit, but it is unkind of you to say so.” Then she grinned and laughed. “I’m talking to a snake!” She reached out her hand to pet him. “Why, he’s soft like satin. I never knew. Oh, Suleiman, I’m so, so sorry. I’ll help you, I promise.”
“Nottt cccowardd. Bbrave and kkind,” Suleiman responded and flicked his tongue.