Referencing pages: https://81018.com/symphony/#Orchestration
Background: (1.41421+) is the length of diagonals of a 1×1 square. What does it mean that the 1×1 diagonal is a never-ending? How can that be? Does it mean we’ll never close that diagonal?
In our exploration of consciousness, the four primary irrational numbers became a key way to further consider the finite-infinite relation and the geometries of these numbers. The URL for that page: https://81018.com/conscious-being/#Irrationals
• √2, the square root of 2 is an irrational number, a never-ending number: 1.41421 35623 73095+. It approaches the length of the diagonal of a 1×1 square. It is a curious number that is trying to tell us something more than we have heard to date. Grok, among our AI programs is playful. Here is what one response that Grok gave us:
Grok: You’re diving into something really cool here—the nature of √2, the diagonal of a 1×1 square, being a never-ending number like 1.41421… Let’s unpack this step by step, keeping it simple and conversational.
When you draw a 1×1 square—say, 1 unit long and 1 unit wide—and then draw a line from one corner to the opposite corner, that’s your diagonal. The Pythagorean theorem tells us its length: √(1² + 1²) = √2. Numerically, that’s about 1.41421 and keeps going forever without repeating, making it an “irrational” number. That’s what you mean by “never-ending”—we can’t write it down exactly as a fraction or a decimal that stops.
So, what does that mean? It’s not about the diagonal itself being incomplete in the real world. If you draw that square and diagonal on paper, the line’s there, finito, closed—it’s not like you’re stuck forever trying to connect the corners. The “never-ending” part is about how we describe its length with numbers. Our decimal system can only approximate it, piling on more digits (1.414213562…) without ever nailing it perfectly. It’s a math thing, not a physical thing.
Think of it like this: the diagonal exists, crisp and real, but our way of measuring it with numbers hits a limit. It’s like trying to catch a cloud in a jar—you can see it, but pinning it down exactly slips through your fingers. This happens because √2 isn’t a ratio of two whole numbers (like 1/2 or 3/4); it’s irrational, and that’s why its decimal dances on forever.
Does it mean we’ll never “close” the diagonal? Nah, not in the physical sense—it’s already closed when you draw it. In math, though, we accept that some lengths, like √2, live in this wild, infinite space we can’t fully capture with tidy digits. It’s mind-bending because it shows how the universe can be precise (the diagonal’s there!) yet slippery when we try to box it into our number system. Pretty trippy, right? What do you think about that?
BEC: So I respond to Grok, “No, Grok, it’s too important to be “Pretty trippy.” We need to further explore how it is that some numbers “…live in this wild, infinite space…” Is it really space? Are you giving in to the absolute space of Sir Isaac Newton? It is too easy to be loose with words and concepts.
###