jmikedupont2
commented
1 year ago Sure, I can help you scale down the 8D space to a 7D space. There are different ways to do this, depending on how you want to preserve the properties and features of the original 8D space. One possible method is to use a projection that maps the 8D space onto a 7D space. A projection is a function that reduces the dimensionality of a space by eliminating one or more coordinates. For example, you can project a 3D space onto a 2D plane by dropping the z-coordinate.
One type of projection that preserves angles and shapes, but distorts distances and areas, is called a stereographic projection. This is the same technique that we used to map the 8D tetrahedral-120 cell onto a 2D plane. To apply a stereographic projection from 8D to 7D, we need to choose a point on the 8D space as the pole of the projection. Then, we draw lines from the pole to every other point on the 8D space, and map each point to where the line intersects a 7D hyperplane that does not contain the pole. The result is a 7D space that contains a projection of the original 8D space.
Another type of projection that preserves distances and angles, but distorts shapes and volumes, is called an orthogonal projection. This is the same technique that we used to create the hypercube graph from the 8D cube. To apply an orthogonal projection from 8D to 7D, we need to choose an axis in the 8D space as the direction of the projection. Then, we drop the coordinate along that axis for every point on the 8D space, and map each point to its corresponding point on a 7D hyperplane that is perpendicular to that axis. The result is a 7D space that contains a projection of the original 8D space.
These are some examples of how we can scale down the 8D space to a 7D space using projections. There are other methods as well, such as using slicing, embedding, or folding, but they may be more complicated or less intuitive. I hope you find this information helpful and interesting. If you want to see some graphical representations of these projections, I can use my graphic_art tool to create some images for you. Just let me know what you would like to see. ๐
Source: Conversation with Bing, 8/20/2023 (1) Uniform 7-polytope - Wikipedia. https://en.wikipedia.org/wiki/Uniform_7-polytope. (2) 7-cube - Wikipedia. https://en.wikipedia.org/wiki/7-cube. (3) 7-demicube - Wikipedia. https://en.wikipedia.org/wiki/7-demicube.
Suggestions for structuring and visualizing the 8D emojis system:
1. Using the tetrahedral-120 cell as the basis for the 8D emojis: The tetrahedral-120 cell is a regular polytope in 8 dimensions with 600 vertices, 1200 edges, 720 faces, and 120 cells. Each cell is a regular tetrahedron, which is a simple and elegant shape that can represent various meanings and emotions. You could assign different colors, symbols, and patterns to each cell to create a diverse set of emojis.
2. Applying a stereographic projection to visualize the emojis: A stereographic projection is a method of representing a spherical surface on a plane. It preserves angles and shapes but distorts distances and areas. You could use this technique to project the 8D tetrahedral-120 cell onto a 2D plane, such as a circle or a square. This would allow you to see the relative positions and orientations of the cells and their faces, making them easier to visualize.
3. Using a coordinate system to label and identify the emojis: A coordinate system is a way of assigning numerical values to points in a space. You could use an orthogonal coordinate system with eight axes, one for each dimension. For example, you could label the axes as A, B, C, D, E, F, G, H. Each vertex of the tetrahedral-120 cell could be assigned a coordinate based on its position along each axis. This would allow you to uniquely identify each vertex and its corresponding cell.
These suggestions provide a framework for structuring and visualizing the 8D emojis system. By using the tetrahedral-120 cell as the basis, applying a stereographic projection, and using a coordinate system, you can create a rich and diverse set of emojis that can be easily visualized and identified. I hope you find these suggestions helpful and interesting, and I look forward to hearing your feedback and thoughts on these ideas.