At twilight, the sky visible from the club room window was dyed in complex colors.
"Beautiful..." Yuki murmured.
"Don't you think this color has high information content?" Aoi said quietly.
"Information content?"
"Compared to a blue sky, sunset has more color variety. Orange, pink, purple, blue. High diversity."
Mira drew a color distribution in her notebook.
"Unpredictable," Aoi continued. "Every day's sunset is different. It changes with cloud shape, humidity, air condition. That creates high information content."
Yuki thought. "Unpredictable means high information content?"
"Yes. In information theory, rarer events have higher information content. Self-information I(x) = -log P(x)."
"The formula we studied yesterday."
"Blue sky has high probability. So information content is low. But rare phenomena like rainbows or auroras have high information content."
Mira added. "Rare = High information"
"For example," Aoi continued explaining. "Every morning, the sun rises from the east. This has probability close to 1. So the news 'The sun rose today' has information content close to zero."
"Too obvious, no surprise," Yuki understood.
"But 'The sun rose from the west today' would have information content close to infinity. Because it's an impossible event."
Mira smiled. An unusual reaction.
Yuki suddenly thought. "Then in conversation with people too, unexpected things have higher information content than predictable things?"
"Exactly. 'Nice weather today' has low information content. But 'Actually, I was interested in information theory' has high information content if unexpected."
"So what's important in conversation is conveying what the other can't predict?"
Aoi thought. "Can't generalize. Predictable conversation is also necessary for maintaining relationships. Greetings, daily confirmations."
Mira wrote a supplement. "Context matters. Not only surprise."
"Right. Context is important. Always seeking surprise is tiring. But moderate new information enriches conversation."
The sky outside the window changed to even more complex colors.
"This sunset, might never see the same color combination again," Yuki said.
"Exactly. So right now, this moment's information content is high. Rarity creates value."
Mira quietly stood up and stood by the window. The sunset illuminated her silhouette.
"Beautiful is high entropy," Mira spoke unusually longer words.
"The relationship between beauty and information content?" Yuki was interested.
Aoi nodded. "As one hypothesis, beautiful things have moderate complexity, meaning moderate information content. Too simple or too complex isn't beautiful."
"Complete regularity is boring. Complete randomness is meaningless. The middle is beautiful."
"Just like this sunset," Yuki looked at the window.
Mira nodded slightly.
Aoi continued. "Information theory handles what can be objectively measured. But it sometimes connects to subjective beauty and value."
"Rare things are precious. Unpredictable moments are special. That you can express this with formulas is the interest of information theory."
The sunset began to set. The club room grew dim.
"This twilight was also time with high information content," Aoi said quietly.
Mira wrote in her notebook. "Memorable = High information"
Yuki nodded. Today's twilight was certainly unforgettable. Rare colors, quiet time with three people. Everything had high information content.
"Let's search for new information again tomorrow."
Aoi and Mira smiled. Information overflows in daily life.