**Making Kaleidoscope to Represent Coxeter Groups. **

**What was our process with this project?**

The project began with the conceptualization phase, where we identified the goal of creating a kaleidoscope to represent the symmetries of Coxeter groups visually. This phase involved extensive research into Coxeter groups to understand their mathematical foundations and how these could be interpreted through a physical model. We then moved into the design phase, where we sketched out the initial designs of the kaleidoscope, focusing on how to arrange mirrors to reflect the symmetries of Coxeter groups accurately.

The assembly process started with the selection of materials. We chose a combination of colorful beads to create the internal patterns, aiming to capture the vibrant and dynamic nature of mathematical symmetries. To construct the body of the kaleidoscope, we opted for sturdy card stock which was rolled up into a tube, inside which we planned to house the mirrors and decorative elements. We carefully cut plastic sheets to serve as the end piece that would hold the beads and glitter in place, ensuring they could move freely when the kaleidoscope was rotated.

For the internal reflective surfaces, we decided against traditional mirrors and instead used reflective paper, aiming for a more cost-effective and safer alternative. This paper was cut and glued onto thick card stock strips that were the same dimensions of the reflective material. Three strips of the now-thick reflective material were taped together to make a triangular prism positioned inside the tube to form the reflective surfaces necessary for creating the patterns. We experimented with different angles and configurations of these surfaces to best represent the Coxeter groups’ symmetries.

The assembly was meticulous, with each component carefully positioned to ensure the kaleidoscope functioned as intended. Once assembled, the kaleidoscope was subjected to multiple tests, rotating it to observe the patterns formed by the reflections of the beads and glitter off the reflective surfaces. These tests were crucial for understanding how well the physical model represented the mathematical concepts it was designed to illustrate**.**

**What Worked Well? **

The assembly phase, particularly the experimentation with mirror angles and configurations within the kaleidoscope, was highly successful. This experimentation allowed us to explore a range of symmetrical patterns, offering valuable insights into the practical representation of Coxeter groups’ symmetries. The choice of colorful beads and glitter as the medium for creating internal patterns proved effective in adding a layer of visual appeal to the mathematical concepts, engaging viewers not just intellectually but aesthetically as well.

The innovative use of reflective paper, despite its limitations, initially seemed promising as a cost-effective and safe alternative to traditional mirrors. This choice allowed us to experiment with the kaleidoscope’s internal design without the risk and expense associated with cutting and installing glass mirrors.

Throughout the project, our iterative design process—designing, testing, and refining—enabled us to make incremental improvements to the kaleidoscope. With each test, we gained a better understanding of how different materials and configurations influenced the clarity and complexity of the symmetrical patterns generated, allowing us to make informed adjustments to better achieve our project’s goals. This process of continuous refinement was instrumental in exploring the potential of the kaleidoscope to serve as an educational tool.

**What Went Wrong Along the Way?**

When making our kaleidoscope, we wanted it to make use of intricate symmetries and designs of certain Coxeter groups. We ran into some unexpected challenges due to the choice of materials.

One of our significant hurdles was our use of beads and glitter to create the kaleidoscope’s internal patterns. While these materials added vibrant color and texture, they were less ideal for forming the precise, geometric patterns we aimed to represent. The irregular shapes and inconsistent sizes of the beads and glitter made it difficult to achieve the exact representation that we envisioned.

Additionally, the plastic material at the end of the kaleidoscope posed its own set of problems. We opted for a plastic sheet, however, this choice restricted the free movement of the beads and glitter within the tube, limiting the variety and clarity of the shapes that appeared as we rotated the kaleidoscope. Going forward, we should pay more attention to the balance between material functionality and the physics that govern the kaleidoscope and the shapes that are created.

Another impactful decision was our substitution of reflective paper for mirrors for the interior walls of the kaleidoscope. We noticed that the material significantly diminished the reflectivity and clarity of the images. The loss in luminosity and sharpness of reflected light led to a muted visual.

Going forward, we should pay closer attention to the materials that we use to construct our projects because representing mathematical theories through art is more precise than we were expecting. In order to get the results we envisioned, more precision will be required.

**What was the Mathematics behind this project?**

The mathematics behind this project involves reflections, mirrors, and symmetries, particularly as they relate to Coxeter groups. These concepts are grounded in geometric and algebraic principles that help us to understand and classify the patterns and symmetries we see.

What are Coxeter Groups?

Coxeter groups are mathematical entities that generalize the notion of symmetry in geometric spaces. They are defined by generators corresponding to reflections across hyperplanes in any number of dimensions, and the relations between these reflections are determined by the angles between the corresponding mirrors. We can explore how different geometrical properties and patterns arise from these reflections.

Exploring Reflections

In order to gain an intuition of how reflections work with the mirrors in a given space, imagine you’re in a room where every wall is a mirror. If you were to look in one direction, you can see your own reflection. But what is more interesting is how these reflections can interact with each other, creating patterns of images that seem to extend indefinitely. This is the basic idea behind reflections in mathematics, but instead of a room, we deal with spaces of any number of dimensions.

In the case of the kaleidoscope we built, we’re mostly concerned with the two-dimensional spaces (the surface of the mirrors individually) and three dimensional spaces (the interior of the kaleidoscope)

How Can We Model the Mirrors Mathematically?

We can represent each mirror by a line in two dimensions, or a plane in three dimensions, through which a reflection happens. The formula given, v = 0, tells us how to find the mirror’s position in space. The vector is perpendicular to the mirror, and any point v on the mirror satisfies this equation, meaning it’s exactly on the reflective surface.

When an object is reflected in a mirror, it flips over to the opposite side. The formula Rv = v – 2( v) gives us a precise way to calculate where the reflected image of any point will be.

How Does This Relate to Coxeter Groups?

The way that we chose to arrange our mirrors can be represented as generators of a Coxeter group. The relationship between these generators is defined by the angles at which the mirrors intersect. These properties dictate the types of symmetries that can occur in the given space. Coxeter groups can be classified into finite and infinite types, depending on whether they generate a finite or infinite number of reflections. In the case of this project, we are exploring a physical manifestation of a finite Coxeter group, where the mirrors form a closed system that generates a finite but complex pattern of reflections. For example, if we decided to make sure all the mirrors form angles of 60 degrees with each other, they correspond to a Coxeter diagram of type A_{3}**, **indicating a certain structure of reflectional and rotational symmetries in the generated patterns.

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