Exploring Mathematical Ratios

"Sections Beyond Golden" from the document "Bridges: Mathematical Connections in Art, Music, and Science" by Peter Steinbach delves into the intricate relationships between numerical ratios and proportions, and their applications in art, music, and science. Here are the key points covered in the document:

Understanding the Golden Ratio: The article begins by clarifying the concepts of the golden ratio, mean, section, and proportion. It explains how these ratios have been used historically in art and design, providing a mathematical and historical context.
New Numerical Relations: Steinbach introduces three newly discovered numerical relations that, along with the golden ratio, form a 'golden family'. These relations are not fully understood yet but are compelling in their mathematical and design implications.
Optimal Proportional Sections: The document explores how proportions can be created by multiple cuts of a segment, leading to optimal proportional sections. The heptagon's diagonals, for example, form a 3-by-3 proportion, which is described as the unique optimally proportional trisection.
Mathematical Properties and Applications: The newly discovered ratios have properties that are significant for artists, allowing for easy arrangement of similar figures or dissecting a figure into a set of similar figures. This ties back to the Greek idea of geometric progression, where multiplication can be accomplished by addition.
Complex Proportional Systems: The article discusses complex systems of proportions, such as quasiperiodic sequences and their applications in design and art. These systems offer a rich field for exploration and have practical implications for design and architecture.
Rational Approximants and Fibonacci Sequences: Connections are made between the golden ratio, new ratios, and Fibonacci sequences. The article illustrates how powers of these ratios can be expressed as linear combinations with Fibonacci numbers, leading to a deeper understanding of their mathematical significance.
Future Exploration and Applications: Steinbach concludes with a call for further exploration of these ratios and their applications. He points out that the mathematical world is constantly discovering new phenomena related to these ratios, which could have significant implications in various fields.

Overall, the document provides a comprehensive look at the interplay between mathematics and design, particularly focusing on how certain numerical ratios and proportions have been and can be used to create aesthetically pleasing and harmoniously balanced works in various disciplines.