**In Praise of Mathematics, Reality, and the Power of 3D**
Mathematics is often misunderstood. Many people think of it solely as numbers, arithmetic, and rigid formulas. But what if we looked at math differently? What if we saw it as a tool to understand our world, to bring shape and depth to our experiences, and to reveal hidden layers of reality? In this post, we explore the connection between mathematics, our perception of reality, and the power of 3D modeling—inspired by recent conversations and insights.
### Mathematics as the Study of Abstraction
Most people struggle to define what mathematics truly is. Often, they simplify it to "numbers." But this is just one small piece of the puzzle. Mathematics is the study of abstraction—the language we use to make sense of everything from geometry and motion to complex systems and relationships. It allows us to describe patterns, measure the immeasurable, and explore ideas beyond our immediate perception.
Take, for example, the concept of **edge detection** in an image. It’s a mathematical process that brings clarity and dimension to what we see—like when you observe where a mountain and the sky touch on a horizon. By identifying boundaries, we create a meaningful representation of objects, surfaces, and the space they occupy. Mathematics, in this way, reveals the underlying structure of the visual world, one edge at a time.
### Reality Through the Lens of Edge Detection
In image processing, **edge detection** helps bring out significant features in a scene—the points where reality transitions from one element to another. Imagine these edges as zippers, threading together different regions of an image. These zippers close gaps, connect pieces, and form a coherent picture of what we see.
As we move between frames of a movie, these zippers help map color consistency and visual details across scenes, adding depth and context. Consistency in edges and colors creates a kind of visual signature—a fingerprint that represents the scene. This process allows us not only to create beautiful 3D models from flat images but also to understand how changing perspectives influence what we see and how we perceive continuity in a story.
### The Power of 3D: Bringing Abstraction to Life
With tools like **DaVinci Resolve**, **Mathematica**, and **Unreal Engine**, we can take abstract mathematical concepts like edge detection and image analysis and translate them into something tangible. Imagine taking footage from a movie, enhancing it to make details crystal clear, and using that footage to create a 3D model that can be explored in virtual reality. Suddenly, something fleeting—a moving image—becomes an immersive, multi-dimensional experience.
The power of 3D lies in its ability to transform the abstract into the real. It’s about creating something you can walk around, inspect from different angles, and refine by interacting with it. Each frame, each edge, and each color difference adds layers of information that bring us closer to understanding what we’re seeing. Reality becomes richer when viewed from multiple perspectives, each layer revealing a little more of the story.
### Threads, Zippers, and the Art of Perception
Considering the edges of an image as zippers is a beautiful metaphor. These zippers connect different parts of our visual world, allowing us to move seamlessly from one element to the next. This idea also reflects how we, as observers, construct our understanding of reality—threading together pieces of information to create a unified whole.
Mathematics, with all its abstraction, gives us the tools to bring these threads together, close the gaps, and see the world with greater clarity. When we apply mathematical processes like edge detection and 3D modeling, we are not just performing technical tasks; we are engaging in an ancient human endeavor to understand, connect, and celebrate the world around us.
### In Praise of Curiosity and Exploration
At Kaivala, we celebrate the joy of exploration—whether it’s exploring the natural world, understanding the abstract world of mathematics, or combining the two through technology. We believe that by taking the time to observe, re-spect (observe again), and refine, we can reveal something extraordinary about the world. This is why we use tools like edge detection and 3D modeling: not just to create beautiful images but to bring people closer to a deeper understanding of reality.
The next time you look at a photo, consider the edges. Think of how they form a zipper that stitches together different parts of what you see, bringing a flat surface to life. There is so much more beneath the surface, and with mathematics, curiosity, and technology, we have the power to reveal it.
### Mapping Bins to Camera Angles
Another important concept is how bins in an image can be mapped to the camera's angle of observation, especially when using a fixed lens. Imagine dividing an image into different sections or 'bins'—each of these bins represents a specific portion of the scene as captured through the camera lens. For a fixed lens camera, each bin can be directly correlated to a unique angle in both horizontal (theta) and vertical (phi) directions.
This allows us to precisely determine which part of the scene each pixel represents based on its position within the image and the camera's known field of view. By mapping these bins to specific angles, we create a geometrical understanding of what the camera sees, which is incredibly useful for reconstructing scenes in 3D. This mapping translates flat images into spatial data, enabling us to create more accurate and contextually rich models of reality.
### Talking About Size
Finally, we want to talk about size. We save this calculation for last because keeping the values less than one makes it easier to check the work. By working with values below one, we ensure that our calculations are manageable and allow for quick verification, especially when mapping these bins and angles in a 3D reconstruction. As well this algorithm does not need to take in consideration all of the images of your movie. You could most simply take a 2x2 array and take the beginning middle and the end, and have a very abstracted perspective of your movie.