When the sun had barely begun its path across the sky the next morning, the girls raced to the Palace of the Partal where the Sufi stayed. Tahirah stood at its entrance, smiling at the sight of the hound trotting happily beside them. He waved a found stick around in delight.
“This is a splendid stick!” He dropped it in front of them. “You know, I wouldn’t mind if you threw it.” He waited, ready to chase. While both girls were pleased with this new and cheerier Suleiman, neither wanted to throw the now drool-covered thing. Suleiman finally grinned and grabbed it again. “Still, it’s a great day and a wonderful stick.”
He carried it high in the air, tossing his head in pleasure until he spied Tahirah. He dropped the stick. “There she is. I can’t believe it. What a thrill.” He raced full out toward her, leaping and dancing around her until, to his delight, she laughed out loud.
“I missed you so.” He danced around in a circle once again and ran back to get the stick. “Did you see my stick? I found it on the way here. Isn’t it a great stick?”
Tahirah reached for the stick and, after a short playful tussle with Suleiman, threw it with a practiced arm. Suleiman bounded after it. Ara and Layla hugged Tahirah, then stepped back when Suleiman returned bearing the stick.
Ara watched his antics with mixed emotions. “I’ve never seen Suleiman this happy.” Under the circumstances, it didn’t seem right that her stodgy tutor should be so lighthearted, even if he was magically transformed into a dog’s body.
“It is wonderful, is it not?” Tahirah threw the stick again. “There’s a lesson to be learned from dogs. Life is fleeting. Enjoy it to its fullest.”
Layla nodded. “That’s what the lions said.”
Tahirah stopped. “You spoke to the lions?”
Suleiman returned again with his stick, dropped it at Tahirah’s feet and panted. “No. I spoke to the lions. Or,”—he looked confused—“the cat did, anyway.”
“I think we need to go into my rooms for a more private conversation.” Tahirah said, then led the way. Servants nodded politely with only a few raised eyebrows at the hound trotting happily alongside.
A few moments later, enjoying tea and flaky sweets, they lounged on the cushions. Suleiman stood at Tahirah’s feet, catching pitched morsels of food.
“So, how did you come to speak to the lions?” she asked.
His ears went down, and he whined slightly. “I went to confer with them. They were disdainful of me.” He thought a bit. “Of course, I too would be disdainful of a small cat were I a grand one. Perhaps that was the real problem.”
Tahirah hid her smile. “Perhaps, but let’s explore some other avenues. Think back. What did the lions say to you?”
Suleiman sat down and scratched his ear before looking at her plate, his head tilted. “I told Ara and Layla already.” He glanced at Tahirah. “I would remember better if you gave me one more of those little pastries.”
Her lips twitched as if amused but she replied in an uncompromising tone. “No blackmail, thank you very much. If you don’t choose to tell me, the girls will be happy to do so. I wanted to give you the chance to tell your story yourself.”
Suleiman’s hung his head. “I’m sorry. You’re my pack. Of course I’ll tell you. I don’t really need the treats.” He turned his head slightly from the dish of pastries, drool dripping down his jaw.
Tahirah sighed and shook her head before tossing him a treat. “Suleiman, here, have a pastry. I can’t stand to watch a grown dog drool.”
It disappeared in one snap, and with a muffled bismillah, Suleiman gulped it down. “The lions were miffed at the cat. Discipline and Reason growled at him.”
“You mean YOU,” Ara interrupted. “You were the cat.”
“Perhaps in the strictest sense, that may have been true,” he conceded, getting up to circle uncomfortably. “Anyway, they, the lions, wanted me to use my enchanted time better. To learn and understand from each transformation.”
“And are you?” Tahirah questioned.
He sat down and studied his paws. “As a lizard, I learned how it feels to be powerless. As a snake, I relearned the value of life. In mouse form, I understood that size does not define spirit.” He whined again before speaking. “The cat learned that not all things can be done alone.” He wagged his tail and perked up his ears. “But dogs, dogs are great. I can see this. Allah has, in his ultimate wisdom, given us the best traits.”
“Which are…” Tahirah encouraged.
“Isn’t it obvious? We’re loyal to our pack and brave and joyous. A true companion. Best at work, best at—”
“Best at modesty,” Ara said wryly, “most humble, most…”
“You are picking on me.” Suleiman tucked his tail under his rump, his ears drooping once again. “Don’t you like me anymore?”
Instantly ashamed, Ara leaned over and patted him on the head. “You are perfect. Whatever shape you take, I love you. I’m sorry. Everything seems to be happening too fast. I’m hiding things from Father, and I’m scared.”
Suleiman grinned and gave her a couple of quick, understanding licks.
Tahirah reached out and put her arm around both friends. “Speaking of which, we need to complete the next lesson. Suleiman, would you like to teach this one? I think rotation would be best.”
He barked twice, then looked chagrined. “Sorry, when you speak many languages, it’s hard not to fall into careless habits. Actually, I would like to explain rotation.” He looked around for an example, and in turning, discovered to his wonder, his tail. He continued circling, trying to grab it. “See, here is a perfect example.”
The girls saw nothing but a silly hound chasing his tail. “Perfect example of what?” Layla asked.
He continued the pursuit, making an occasional half-hearted grab for it. “Rotation. I’m turning around. It’s obvious.”
Ara turned a cartwheel while Layla did a pirouette. “You mean like that?”
Grinning, he sat. “Yes, but now you have to look at it in slow motion to see how it’s different from a vertical reflection. Tahirah, would you pull out the tiles again?”
She drew out four tiles and placed them before the dog. Suleiman placed a paw on the first tile. “Each of these is identical, like a translation, right?”
“Yes, “ Ara agreed, “just as before.”
Suleiman wagged his tail and woofed. “But now we will create rotations. We take the first one and place it so.” He nudged two tiles into place. “See how with the first tile, the blue part of the design is on the top? And then how the second is turned one hundred eighty degrees? It is rotated.” He placed a paw dead center between the two tiles. “With rotation, you pretend there is a point or hub in the center around which the object moves so that it appears to have spun.
Now you try it.”
Ara looked at the tiles with interest. This was a lot like a game! She turned the next tile one hundred eighty degrees and placed it beside the first two.
Suleiman lay down beside them. “Pretend that you’re holding your paw—sorry, I mean finger—in the middle. An immovable point. That point, at your finger, must stay in place as you turn the tile around.”
“This is the third of the four motions,” Tahirah said, joining in. “Reflection—which we also call a flip—translation or slide and now rotation,” she finished, having ticked them off on her fingers. “One of the remarkable things about rotation is that double reflections also rotate.”
Ara looked up. “Because it looks the same upside down and right side up?”
“Exactly. It includes a one hundred eighty degree rotation in the design, so it looks the same upside down or right side up.” Tahirah turned both tiles upside down and then back again.
The dog woofed. “You forgot to mention that there cannot be a flip in a rotation.”
The mathemagician smiled, patting his head in agreement. “That is correct. For it to be considered rotation, it can’t have a mirror in the design. It does not reflect. If it did, it could be a double reflection or possibly the seventh band symmetry which you have not yet learned.”
Layla asked. “Could we see another example?”
“Take these same tiles and make another rotational symmetry,” Tahirah suggested.
Uncertain, Layla picked up a tile. “How do you start?”
Suleiman woofed encouragingly before stuffing his nose under her arm. “You can do this. Look at all you’ve learned so far. Try.”
She placed the first piece with the blue part facing down. The next tile she lined up exactly the same, then turned it one hundred eighty degrees to the right. Encouraged, she took the third piece and, placing it the same way as the second tile, rotated it one hundred eighty degrees yet again. The last tile she turned one hundred eighty degrees until the design cartwheeled across the floor.
Suleiman barked in glee and ran around the room. “See, you did it!”
Layla grinned. “I guess I did.”
Everyone jumped at the knock on the door. Tahirah called out permission to enter. Her servant did so, then, bowing near to the floor, handed her a card on a silver tray. After dismissing her, Tahirah read the card. “The sultan requests the presence of his daughter and niece immediately in the Hall of the Kings. He wishes the dog to be brought before him.”
She looked at the children, her eyes dark with concern. “It seems the owner of the dog has come forward.”