“That’s terrific. I know you’re excited and busy, and I won’t keep you long…Next Tuesday, I’m giving a lecture at the Library Society. The topic is ‘Beauty and Mathematics.’ Since you’re an artist, I thought you might enjoy it, and I was wondering if you’d like to go with me. The lecture will last about an hour, ninety minutes with questions, and there will be refreshments afterwards.”
Beauty and mathematics? In what world did they invent such off-the-wall topics?
In her mind, Karen was back in Ms. Thomas’s algebra class, third seat on the first row, her eyes focused on a chalkboard covered with equations.
She imagined Ms. Thomas’s voice. It’s obvious, then, that X equals twenty-four-point-two-five and therefore Y is the most beautiful of them all.
Right. Math had never been beautiful. Not to Karen.
“Well…uh…” She started to beg off. Tuesdays were always busy, staff meeting all morning, the exhibit still in full swing, and…
She took a deep breath. Ninety minutes and refreshments. Their second date down, and the chance to find out more about Mark Stuart. She could do that.
“If you’re busy, it’s all right…”
“No. I’m not busy that night. Not at all. No. Sounds like a terrific idea. I’d love to go.” Her voice was as flat as the line on Dr. Burns’ graph, and she heard Mark chuckle as if he knew she was not being truthful.
As she placed the telephone on her desk, Vicky knocked at the door.
“Way to go. Looks like a winner. We had only two,” she held up two fingers, “two copies of the catalog left when we opened this morning and one of those went while we were in our meeting. I’m having two cases sent up from the warehouse.”
“That’s great” Karen exclaimed. “I’m so happy.”
“Speaking of winners,” Vicky perched on Karen’s desk. “How was Mark Stuart? Have your own little party when he took you home?”
Karen rolled her eyes. “Went to bed when I got home…Alone,” she snapped as she saw Vicky’s eyes sparkle. “Not that I would have anyway, but I can no more conceive of Mark Stuart’s suggesting we sleep together than…than I can imagine walking naked through Marion Square at lunchtime with my mother and her bridge club standing on the curb watching.”
“What’s wrong? Things seemed to be going so well. Did you not have a good time?”
“I did have a good time, an extremely good time. It was the best date I’ve had in years.”
“You let him kiss you good night?”
Karen crossed her arms over her chest.
“That’s just it…nothing happened. He made not the slightest move to kiss me. That’s the story of the evening. We saw the exhibit. Mark talked about one set of paintings as if he were an art professor. He seems to be very nice, he’s very polite, I really enjoyed talking to him, and…and…He didn’t laugh. He didn’t smile. I know no more about him than I did when he rang my doorbell on Friday night.” She threw her hands up in surrender. “I’ve no idea what interests him, but I’m pretty sure it’s not me.”
“Mark’s hard to read,” Vicky agreed. “What about your second date?”
Karen turned on her sharply. “Second date? Humph. We’d been talking when you knocked. I chose to go to the gala, so the second date was his choice. He chose…” she rolled her eyes dramatically, “a lecture he’s giving at the Library Society next Tuesday. The title is ‘Beauty and Mathematics.’ Can you believe that? Is he truly going to try to reduce beauty to a set of numbers or to a formula?” She shook her head. “Honestly.”
“Actually, it sounds interesting.”
“No way.”
Vicky laughed. “Okay, maybe not that interesting, but I will tell you Mark has the reputation of being a very good teacher. If it’s possible to breathe life into that topic, I’m betting he can do it.” She stood. “I need to go. Talk with you later.”
“Mathematicians reduce everything to numbers, don’t they?” Karen said, shaking her head. “Probably has a formula for a good kiss,” she mumbled as Vicky reached the door.
“Ask him,” Vicky called over her shoulder. “It would be a good lead-in when he takes you home.”
Karen heard Vicky laughing as she walked away, but she still very much doubted that their second date would end any differently than had their first.
***
As they arrived, Karen had chosen a seat toward the back of the room, while Mark had taken his tablet and a folder containing his notes to the lectern at the front. She watched as he arranged his notes, placing them in several stacks across the table. He opened his tablet, entering an address to link it with the projector that hung from the ceiling. Images began to pop on and off the screen behind him as he checked the connection.
Karen looked over her shoulder, surveying the hall. Every chair was taken, and the walls were lined with people who were standing. She had expected a handful of geeky students, but she guessed there were at least two hundred people packed into the small room at the Library Society building. A few might have been college age, but most appeared to be older.
She looked closely, recognizing a few of them. Mr. Bannister was her father’s attorney. Dr. Nelson, whose son she had once dated, taught at the medical college. She waved to two of her mother’s friends.
She understood paying to attend a lecture on some aspect of beauty, but why would they spend good money to hear someone talk about mathematics? She shivered. If Mark had not invited her to attend as their second date, she would be far, far away from this place. She imagined herself curled up on her sofa, a cup of hot chocolate in her hand, engrossed in the latest episode of Grey’s Anatomy. She suspected Mark had planned this date as a way to encourage her to bail prior to date number three. That would leave her on the hook with her mother and him in the clear. Then, Karen reminded herself not to be such a cynic.
As the space began to fill, the noise level rising, Mark joined Karen at the back of the room. He leaned close so she could hear him speaking above the din of voices. “I always tell my students that I like the illusion they are hanging on every word I say.”
“I’m…I’m just…” Karen looked up, confused. What did he mean by that?
His eyes sparkled. “So I recommend seats toward the back of the room if they plan to use their cell phones…You’ve chosen well.”
He placed his hand on her shoulder and gave a light squeeze before returning to the front of the room, greeting the woman who would introduce him.
Karen glanced down at the device in her left hand, and she could feel herself blushing. Had he been teasing her? That little half-smile. He squeezed her shoulder…Or was he angry? Impossible to tell.
She certainly had not planned to return to the game she had been playing when Mark had arrived at her apartment that evening, even if she had been on the cusp of a personal high score when the doorbell rang. Of course, when he had walked over, she was holding her phone…
She stuffed her cell deep into the tote bag she’d placed on the floor beside her chair and sat straight, her eyes fixed on Mark, sighing deeply.
“It’s just for one hour,” she whispered.
***
The group grew quiet as the president of the society introduced Mark, and they clapped loudly as he stepped behind the podium. Karen noticed several in the audience lean forward, as if they wanted to hear every word of Mark’s lecture.
“I have friends who would tell you the title of my lecture is an oxymoron,” he began, “a contradiction, nonsensical. Beauty and mathematics. Opposites.” His eyes cut quickly to Karen, then he looked away.
“Mathematics, they would tell you, is intellectual. It is objective and unfeeling, black and white, right and wrong. If someone tells you one plus one equals three, then that someone is wrong.” He taped his fist on the lectern for emphasis.
“Beauty, on the other hand, is emotional and personal. Beauty deals in shades of gray. It is subjective and, when you tell me that a woman or a painting or a piece of architecture
is beautiful and I say it is not, then neither of us is wrong. We are simply different.” He raised his hands in a gesture of acceptance.
“Mathematicians are cold, machinelike, interested only in the bottom line, the one correct answer.” He glanced at Karen, smiling. “And they don’t drive sports cars.” The man sitting next to Karen chuckled.
“Artists are warm and lovable, human, able to accept and to live with multiple answers and absolute contradictions.
“Beauty is not mathematical, and mathematics has no room for beauty. The two are at opposite ends of some pole and, as the expression goes, never the twain shall meet.”
He paused, looking across the room, catching the eyes of person after person.
“My friends are mistaken…Not about mathematicians. We are cold, machinelike, and unfeeling, although some of us do drive sports cars.” He winked and smiled, and the audience laughed. “They are mistaken in their belief that beauty and art are not mathematical. Tonight we are going to be talking about a basic principle of artistic design, a mathematical principle, the golden ratio.” He pressed a key to activate the projector.
“Look for a moment at this slide. What do you see?”
After a moment of silence, Karen heard a woman’s voice. “Five rectangles?”
“Of course.” Mark nodded. “You thought it was a trick question, didn’t you?” The members of the audience chuckled.
“Five rectangles. Look at them…Which do you like the best?”
Someone laughed.
“No, seriously. Is one of them prettier, or more well balanced, or more attractive in some manner than are the others? Does one of them hold your attention better than others? Look for a moment. I won’t ask you to explain your answer, but we are going to take a vote.”
“Rectangle number two is a weird-looking square,” a man standing by the rear wall called out. “It’s not a rectangle.”
“Mathematicians define a square as a rectangle whose four sides have equal lengths. In this case, though, the length is slightly, just slightly, longer than the height.” He smiled. “Perhaps that small difference is why you call it weird.”
Mark paused for a moment. Then he pointed at the first rectangle. “Who thinks this is most attractive? Raise your hands.”
He counted three hands.
“How about this one?” He paused.
“No one likes the weird square?”
People laughed.
He repeated the process for all five rectangles.
“So, gosh, most of you…I’d say just about seventy-five or eighty percent of you, see the fourth rectangle as being most attractive in some sense.” He paused for a beat. “Congratulations. You are correct.”
Again, the audience laughed.
“If you measure the fourth rectangle and divide the length by the height, you’ll calculate a ratio of about one point six. This is the golden ratio, sometimes referred to as the golden mean, the golden number, or the divine ratio. A rectangle that reflects the golden ratio is a golden rectangle.” He smiled. “I like divine ratio and divine rectangle because, as we’ll see, even God uses this ratio when he creates beauty.”
Karen stared at Mark. Smiling, laughing, joking with his audience, he barely resembled the man who had asked her for a date.
“Humans tend to see objects that reflect the golden ratio as being more beautiful than objects with other proportions, and we find the golden ratio popping up whenever we see beautiful objects, either natural or manmade.”
He clicked on the next slide, which bore a Greek symbol. “Mathematicians like to name things using Greek letters, and we refer to the golden ratio as phi. Not to be confused with pi, which, while also an irrational number represents the ratio of the circumference to the diameter of a circle—or my Aunt Matilda’s peach cobbler. The ‘rule of phi’ is a technique used for composing pictures.”
He projected a large rectangle on the screen.
“Imagine this is a painter’s canvas or, perhaps, the viewfinder in your camera. Now imagine that two vertical lines and two horizontal lines are drawn across it, resembling a tic-tac-toe board.”
Four blue lines appeared inside the canvas.
“The horizontal lines are sixty percent of the way between the top and the bottom of the canvas, the higher line sixty percent of the way from the bottom and the lower line sixty percent of the way from the top.” He pointed at each line. “The vertical lines are drawn sixty percent of the way across the canvas, one from the left side, the other from the right.
“Now, when these lines cross, they create a number of rectangles. Four of the larger rectangles are golden.” He clicked to the next slide and the four rectangles were outlined in different colors—blue, red, green, and gold. “These rectangles begin in the four corners of the canvas, they overlap, and the very center of the canvas is contained in all four.” He pointed to a small rectangle formed by the overlap of the four large ones.
“In a good composition, the artist places the objects he wants to be most noticed on one of these four lines. The corners of these rectangles, the ones closest to the center, where the vertical and horizontal lines cross,” he pointed to each corner in turn, “are even better locations. In general, the most pleasing images have the center of interest on one of these four corners.”
He’s talking about the rule of thirds, Karen thought. It was a basic rule of composition, one of the first things she’d learned when she began to paint. She’d thought of it as a psychological principle, but, of course, it was mathematical.
“We see phi used in art, in architecture, wherever people want to create beauty.” He looked at his laptop, seeming to be searching for a slide. He glanced toward Karen, as he projected an image of Rouen Cathedral, shrouded in fog.
“This is a painting by Claude Monet, part of his series of Rouen Cathedral. The focus is on the north tower.” He ran his pointer along the tower. “As you can see, Monet placed it on the vertical line on the left side of his canvas, a perfect use of the rule of phi.”
Karen smiled broadly, realizing he’d included the image just for her.
He described how phi was reflected in Da Vinci’s paintings and in the design of the Parthenon and the pyramids.
“Whenever we find something that is beautiful, we tend to find the golden ratio, even in the faces of beautiful people.” He projected a woman’s face on the screen. “I told you that even God uses the golden ratio.”
“That’s…that’s…that actress…Amanda Baker,” a man in the back called out. “She’s gorgeous.”
“Better believe it,” another man said.
“Well, let’s see.” Mark smiled.
With his pointer, he traced the outline of the woman’s middle top teeth.
“When these two front teeth form a golden rectangle, then the person has a perfect smile.”
Mark described various proportions involving the nose, the length of one’s chin, and the position of one’s eyes that reflect the ratio in beautiful faces. Projecting lines across Amanda Baker’s face, he illustrated how well its structure corresponded with those proportions.
“On a scale of one to ten, with ten being the most beautiful, Amanda Baker scores nine-point-three,” Mark said.
“Is that all? No way,” the man sitting next to Karen said.
Mark smiled. “Numbers don’t lie.”
Everyone laughed.
“Actually, nine-point-three is a very high score. Few score above nine, and most of us score between three and seven.” He projected a full-body image of the actress.
“A golden rectangle with its top across one’s shoulders and its base at the navel describes a perfectly proportioned upper body. For the lower body, the rectangle is drawn with the person’s navel at the top and the feet forming the base.” A full-length image of the actress popped onto the screen, with purple rectangles drawn over her body.
“Perfect,” Mark exclaimed.
“One study found these perceptions of beauty to be the
same for all races, across cultures, and during different periods in history. This analysis,” he gestured toward the screen, “was performed with an online program. You upload your photograph, it analyzes the image, and it prints out the results.”
Karen realized she was sitting at attention on the edge of her chair, her entire body tense. This is so cool, she thought. She would have to find that program.
“The golden ratio is based on a sequence of numbers known as the Fibonacci sequence…”
A woman, a college student, Karen thought, turned to the man sitting between the two of them. “I’m only here because my boyfriend was interested.” She gestured to a guy on her other side who held his hand toward the girl, asking for quiet, as if even her whisper might cause him to miss something Mark might say.
“Totally engrossed,” the girl whispered. She paused, watching her boyfriend as he raised his hand to ask a question. “I’m going to find that program, myself.”
Mark described other examples of phi, illustrating how swirls found in nature—snail shells, for example, parts of a pinecone, tornadoes—all reflect the golden ratio. When he finished, Karen led the standing ovation.
After speaking with what seemed like everyone in the audience, answering their questions and thanking them for coming, Mark finally slipped away to meet Karen at the refreshment table, practically seizing a cup of fruit punch.
“My throat is so dry,” he said. “I once had three classes back to back and I could hardly speak by the end of the third one.”
“I really enjoyed your presentation,” Karen told him.
He raised his eyebrows questioningly.
“Didn’t touch my phone the entire time.” She raised her right hand.
Mark nodded. “That’s a compliment.”
For a moment neither spoke, and, as Karen glanced around the room, searching for something to say, she heard music. Although the audience seemed to be drawn from all age groups, the music was primarily “golden oldies,” not quite elevator music, but “safe.” It would offend no one.
As she listened, she heard the soft notes of “Leaving on a Jet Plane,” an old John Denver song, and she softly sang along for a couple of lines.
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