The Principle & Science of Analogy

Introduction to 4th-Dimension and Analogies of Mathematics

An analogy is a comparison between two things or ideas based on their structure or concept for the purpose of explanation or clarification. It highlights the similarities between two different phenomena, concepts or things, often making the unknown familiar. The essence of an analogy is to use a known concept to explain or infer an unknown concept, bridging gaps in understanding. Before diving deep into the math and possible sciences that exists within the fourth dimension, I find it important to discuss how we define our reality by abstract concepts to connect to physical quantities or bodies.

Analogies operate on the principle that if two things (A and B) are alike in one way, they are alike in other ways as well. However, it's essential to understand that while analogies highlight similarities, they do not imply identity. That is, just because two things are analogous does not mean they are the same in every aspect.

3D Cube and Hypercube Analogy

To grasp higher-dimensional spaces, which are challenging to visualize, we often use analogies based on our three-dimensional experience.

3D Cube

- Exists in our familiar three-dimensional reality.

- Defined by length, width, and height.

- Can be visualized and even held, e.g., a dice.

Hypercube (or Tesseract)

- A theoretical, four-dimensional geometric shape.

- An extension of the 3D cube into the 4th dimension.

- Cannot be visualized in its entirety in our 3D space. Instead, we see its 'shadow' or 'projection'.

Analogy:

Consider drawing a 3D cube on a 2D piece of paper. The drawing is just a flat, two-dimensional representation or 'shadow' of the actual 3D object. In the same way, when we try to depict a tesseract (4D) in our 3D world, we're seeing a 3D 'shadow' of a 4D shape.

Principle of Analogy in This Context:

The 3D cube and its 2D representation help us understand the relationship between the 4D hypercube (tesseract) and its 3D representation. By understanding how a 3D object (like a cube) is a 'higher-dimensional' version of a 2D object (like a square), we can then extend that understanding to imagine how a 4D object (like a tesseract) is a 'higher-dimensional' version of a 3D object (like a cube). This analogy doesn't let us fully comprehend the 4D reality, but it gives us a foundation to start grasping the concept. With all this said, analogy is the intuitive tool used by our reasoning. In the vast landscape of mathematical problems and solutions, analogies help mathematicians develop an intuition in the field they are trying learn and discover. Analogies often reveal hidden patterns or structures in problems, guiding the way to solutions.