Herb and Stan met at the coffee shop at 4:30 the next afternoon. The shop was on Congress Avenue and had several tables sitting outside under a couple of small city-beautification trees planted on a widened, brick-covered part of the sidewalk. The coffee shop side of the street was in the shade, and the air was cool enough, barely, for them to sit at the only unoccupied outside table.

Before they sat down, Stan and Herb both briefly noticed a good-looking couple sitting at one of the tables drinking iced coffee drinks. The young woman was reading a book and the young man was reading the Austin Chronicle-or, rather, flipping through it and not reading it. The two had a movie-star look about them, but neither Stan nor Herb was willing to look at them long enough to make a positive celebrity ID.

Across the street sitting in the sun by himself on a green bus-stop-style bench, a middle-aged homeless man in a heavy, dirty, black overcoat, dirty black pants and a big black ski cap smoked a cigar stub, and looked straight ahead, occasionally glancing at the folded newspaper in his lap. Stan and Herb saw the man every time they came downtown. Usually he was sitting on the same bench, but sometimes he was searching through one of the Congress Avenue garbage cans.

“I don’t see how that guy stands the heat of the sun in those heavy clothes,” Stan said as he and Herb sat down.

“I don’t see how that guy stands his own body odor,” Herb replied, just before taking a bite of cranberry bread.

“But have you noticed the cigar smell? He’s always smoking good cigars-maybe that helps negate the body odor,” said Stan, after his first swallow of hot tea.

“Maybe the cigar store owner down the street gives him free samples,” Herb said after swallowing his second sip of coffee and putting the large white china cup back in its saucer.

“Yeh-probably pre-smoked free samples,” said Stan, smiling one of his tight little smiles and squinting his eyes.

“Probably,” Herb agreed, grinning a little and emitting a slight chuckle. Taking a bite of cranberry bread and then a quick sip of coffee, he thought for a moment and then said, “You know, I was reminded of something I couldn’t quite put my finger on when we were talking about your affection for x =1/x last night. I just realized what it is.”

“You mean your affection for x =1/x. I was partial toward x squared equals one, myself. Something related to my unrequited affection for you-know-who, and how it should be obvious there’s only one solution-okay two solutions, negative and positive, but only one number. A song by Three Dog Night comes to mind.” Stan glanced toward the homeless man and took a bite of his walnut-apricot scone then a sip of tea.

“Well,” Herb replied testily, “they’re the same equation.” He paused for a moment and thought about changing the subject and depriving Stan of the thought he’d wanted to share with him. He decided to share it. “I suppose you’ve heard of the golden ratio.”

“Sure, I’ve heard about that a lot,” Stan said, putting his teacup in its saucer and sitting back in his chair. “It’s an eye-pleasing proportionality used by the Greeks in their architecture and also found in nature.”

Pausing and leaning back in his chair like Stan, Herb said, “Yeah, that’s part of the story.” Stan thought Herb was beginning to smirk a little, and indeed Herb looked away from Stan (to hide his smirk, Stan thought) as he continued, “But what is the equation for the golden ratio?” Herb looked back at Stan, waiting for a response.

In no hurry to provide a response, Stan took another bite of his scone, then a sip of his tea. He found the mixture to be very flavorful. “I know there’s a quadratic equation,” he replied after swallowing his tea and scone mixture, “that has a solution that’s a never-repeating decimal expression like 1.618... or something like that.”

“Yep, true enough, and I’m impressed with your memory of the first digits of that decimal number. But the quadratic equation is very much like your beloved x squared equals one, and that is the very cool thing I thought you’d like.”

“Okay,” Stan said, sitting up and taking another sip of tea. After thoughtfully swallowing, he said, “Is the equation x over 1-x or something similar?”

“Very dang close, Stanley old bean,” Herb said, not smirking anymore but actually smiling because he enjoyed quizzing Stan and getting him to think, and he enjoyed the superior feeling he got when he posed a question Stan couldn’t answer quickly. Sitting up straight, and picking up his coffee cup, Herb said just before he took a sip: “But that’s not an equation.”

“Jackass,” Stan said in a good-natured tone of voice. In fact he was happy to be thinking about the connection Herb was trying to make.

“Irrelevant,” Herb immediately said, as if mimicking a judge overruling an objection.

“Well, it’s more like irrevelant, which probably isn’t even a word,” said Stan in response. “But I can come up with an equation.” Stan takes a pen out of his pocket and does some scribbling on his napkin. “Okay, here is the equation,” he said, looking Herb in the eye and sliding the napkin over to him.

Herb took a sip of his coffee, put the cup down, and picked up the napkin Stan slid over to him. On the napkin was written x²/(1-x) = 1.

Herb pulled a pen from his shirt pocket and scribbled on the napkin. “Not quite it, but only off by a minus sign,” he said. “Changing the minus to a plus and kicking the 1+x over to the right-hand side gives the right quadratic equation. But this is what I thought was so cool.” He handed the napkin back to Stan. “Check out the rearrangement I did.”

Stan looked at the scribbled equations on the napkin. The last equation on the napkin, written on the back of the napkin, was x-1 = 1/x.

Sitting up straight in his chair, his back not supported by the back of the chair, Stan held the napkin in front of him and stared at the back of it intently. He kept staring, lost in thought. Herb let him think, without bothering him.

The young couple was getting up from their table, talking to each other now that she had put away her book. Apparently, they had arranged to meet at the coffee shop, and were about to go their separate ways. Herb was suddenly aware, from hearing her voice and seeing her body move in its unique way, that the girl was Rachel, daughter of his ex-girlfriend Denise. Although he had not seen her in five years, he wondered how he could not have recognized her earlier.

“See anything in there about us?” Rachel said as the guy stashed the Austin Chronicle into his army-green knapsack,

“I wasn’t really looking for anything about us,” said the guy, glancing up at Rachel in a distracted sort of way that was nevertheless intense. Herb recognized him then as Jacob Josephson, the movie star son of Jason Josephson, a local film director who had recently made it big. Herb recalled reading, in the very issue of the Chronicle that Jacob had been reading, that a movie with Jacob in it was indeed being filmed in Austin.

“Us?” Herb said quietly to himself.

“What?” Stan’s reverie concerning the equations on the napkin was broken.

Herb leaned over the table toward Stan. “It’s Rachel,” he said in a low whisper, nodding toward the young couple’s table.

Stan turned to look at Rachel as she was shouldering her backpack. For the first time, she looked toward Stan and Herb’s table. At first she looked puzzled, then she smiled and unselfconsciously said, “Hi Herb, hi Stan.”

“Hey, Rachel,” Herb and Stan said simultaneously, causing the three of them to smile briefly. Then Herb added, “How are you?”

“Great. Good to see you guys.” As Rachel spoke, Jacob looked at Stan and Herb, then back at Rachel. Before Jacob could speak, Rachel stepped toward Stan and Herb’s table. “Jacob, these are some friends of mine,”

“How ya doin’” Jacob said, with a friendly nod and a quick, muted smile. He stepped over to Stan and Herb's table with Rachel.

“This is Herb, and this is Stan,” Rachel said, motioning with her left hand toward Herb first and then Stan. Looking at Jacob, she added, “And this is Jacob.”

Being the closest to Jacob, Stan put his hand out. “I thought you looked familiar,” he said to Jacob as they shook hands. “How’s the filming going?”

“Well, we don’t start shooting till tomorrow,” said Jacob. After a brief pause, he added quickly, “At sunrise.”

Herb stood up and shook hands with Jacob across the table. “Good to meet you,” Herb said. Jacob nodded, smiling slightly. Rachel spoke next. “It’s been, like, four or five years since we’ve seen each other.”

“Yep,” Herb said quickly. “Five years. You look really great. How are your mom and Luke doin’?”

“Oh, about the same, doing okay,” Rachel said, moving a loose strand of blond hair out of her face and tucking it behind her ear. “They’re gonna be in the movie, too.”

“So you’re in it, too, then, with Jacob,” Herb said, realizing why Rachel had used the word “us” earlier. “Congratulations on that!”

“Thanks. I’m a little nervous. I’ve been in lots of plays, but never a movie”

“Movies are a lot easier, I promise you,” Jacob said. “Less memorizing.” Then he looked at Herb and Stan and said, “Well, I better get going. Good to meet you guys.” He shook hands with Herb, who was still standing, then with Stan, who was still seated.

“Break a leg!” Stan said as Jacob turned to leave. After he said it, he immediately wondered if it was appropriate to movie making.

“You got it,” Jacob said, turning back around briefly as he walked between tables toward the sidewalk. “You musta seen the script,” he added before turning and walking up the sidewalk toward the Texas state capitol building. Stan, Herb and Rachel all laughed together as Herb sat down in his chair, picked up his coffee cup and leaned back.

Seizing the moment to keep the conversation going, and thinking Herb and Rachel probably felt awkward after not having seen each other for so long, Stan said to Rachel, “Herb and I are exploring the romantic implications of the golden ratio.”

Rachel giggled a comfortable and confident sort of giggle, then asked, “What romantic implications?”

Herb grinned, remembering how good Rachel was at asking direct questions.

“Well, we haven’t figured that out yet,” Herb said, looking at Rachel then at Stan. “The philosopher here is working on a theory.”

“Actually, I just now worked out the theory,” said Stan, looking from Rachel to Herb and raising his eyebrows while smiling a bigger than normal tight-lipped smile. “This equation for the golden ratio, x minus one equals one over x, says when you become an ex, you have to let go of the other person—that's the minus one part—if you are going to turn your life around—that's the reciprocal part. And I re-wrote the reciprocal to make the equation have a more symmetric appearance.”

Stan pushed his scribbled-on napkin to the center of the table where Rachel and Herb could both see it. The final equation on the napkin was x-1 = x^-1.

“Stan, you are a genius! That’s a beautiful equation!” Herb exclaimed as he picked up the napkin. Stan looked up at Rachel, who shifted her weight and adjusted her backpack on her shoulder, smiling at Stan as she did so.

“You guys ought to write a book,” Rachel said, looking back at Herb, “about the algebra of breaking up.”

“If only it were that simple,” said Herb, taking a sip of his coffee but finding it too cold to be enjoyable.

“Well, I better get going,” Rachel said, looking at her watch. “Mom wants me to get some stuff at the store for dinner. It’s going to be one hectic night around our house. With the filming starting tomorrow, everybody’s nerves are jangled a little bit—a little bit more than usual, I mean.”

“Yeh, I’d be nervous too. Congratulations again on getting parts in the movie. Tell your mom and Luke I said hi,” Herb said as he stood up, which prompted Stan to stand up also.

“And tell them hi for me, too,” Stan said.

“I will,” Rachel said as she smiled and hugged Herb. “Good to see you guys.”

“Good to see you,” said Herb, returning Rachel’s smile as they separated and Rachel stepped over to hug Stan.

“Take care,” Stan said as he and Rachel hugged.

“You too,” said Rachel, adding a quick and quiet “bye” as she turned and walked up the sidewalk in the direction Jacob had gone.

“She’s turned out to be a graceful and beautiful young woman,” Stan said as he and Herb sat down.

“She always was,” said Herb, before taking his last bite of cranberry bread, which he washed down with a sip of cold coffee. “Now I need to get going,” he said quickly, looking at his watch. “I’m teaching a class at 6 tonight, you know. But first here’s another way to write the golden ratio equation, which I just thought of after seeing your version." Herb picked up a fresh napkin and began writing on it. "You have x, which is really x to the first power," he said as he briefly glanced up at Stan. "Then you have one, the loneliest number, which can be written as x to the zeroth power, right? Then, lastly, you have x to the minus one power."

Herb pushed the napkin over to Stan, who looked at it and saw: x¹- x⁰- x^-1= 0.

“Well,” Stan said, looking down at the napkin, “I guess I’m even smarter than I thought. But I still like my way of writing it better."

"Well, naturally," said Herb as he stood up and picked up his bookbag. "And it is definitely a beautiful way of writing it. But having dealt with sequences and series in math, I couldn't resist the temptation to write it as consecutive powers of x, like it's a part of an infinite series. Now I gotta go, bro, so take care and get in touch about the thought experiment we've been neglecting today."

"Sorry to say I'm losing interest in it," Stan said, looking in Herb's direction, but looking past him rather than at him. "But, yeah, I'd like to come up with an answer to the original question about the time of the streetlight flash in the two rest frames."

"Let's do that, then," Herb said, starting to walk toward his nearby bicycle.

"Hey!" Stan suddenly said, causing Herb to turn around quickly. Before Herb could say anything, Stan turned to him and said, "Remember, stay away from the Golden Arches."

Herb laughed and said, "I'll do my best." Then he got on his bike that had been leaning against one of the city beautification trees on the sidewalk and rode off.

One of the café staff was cleaning off Jacob and Rachel's table, so Stan figured he would go ahead and leave so his table could be cleaned, and he wouldn't have to think about whether to take his and Herb's cups and saucers inside.

After he'd walked about a block south on Congress he passed the homeless man dressed in dirty black clothes who'd been sitting on the bench across the street earlier. Stan was about to nod and say hi to him, but the man glared so ferociously at him that Stan just looked away.

After walking half a block further, Stan looked back briefly and saw the homeless man had opened the hinged top on one of the green metal trash containers near the café. Then Stan rounded the corner at 8^th and Congress, headed west toward his car a few blocks away. No one was in sight on the sidewalks nearby, so instead of whistling, as he often did, he began to sing the theme from George of the Jungle in a quiet but energetic voice.

(to be continued)

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