In this alternate reality, time and space might behave differently, allowing the possibility of alternate perspectives and alternate paths. This would mean that four dimensions of space-time (3 spatial dimensions, and one time) exist within our observable universe, while the other six dimensions are curled up within subatomic particles.įinally, it is also possible to imagine four (or more) dimensions from an alternate-reality perspective. String theory in physics suggests that our universe is actually made up of 10 dimensions. The second way to think of four dimensions is from a physical perspective. In the physical realm, four dimensions could represent the four fundamental forces of nature: gravity, electro-magnetic, strong and weak nuclear forces. Mathematicians often use higher dimensions to represent abstract concepts and problems that would otherwise be difficult to visualize.įor example, four dimensions can represent the four attributes of an object, such as length, width, height and time. The first way is from a mathematical perspective.
#Tesseract 4d shapes code
We hope others interested in this topic will find the source code useful for constructing more complex visualizations and examples not possible in the app today.It is possible to think of higher dimensions in a few different ways. We draw the former as a wireframe and the latter as a solid. We simply define the 16 points in a tesseract, as well as the connecting edges then we define a cube in 4D space. Being able to follow one face, and to see how it actually is a cube that just gets distorted as the view changes is really helpful.Įven though we have no UI to set this up in the app, it is not difficult to put together with a few lines of code. This 4-dimensional cube appears to distort as it rotates. To me, this is much easier to understand than standard animations of rotating tesseracts. But it's really cool, so I wanted to show it.Ī rotating tesseract with one face highlighted This is not currently possible to do using just the web viewer, so it's not included with the examples above. We would never have gotten this far without all the insightful research we've built upon. Thanks to Nick Nooney for the marching cubes implementation.
Thanks to Professor Paul Humke for spearheading this project and acting as our advisor. The main developers on this project have been Omar Shehata, Joe Peterson, Tianyu Pang and Justin Pacholec. The developer docs provide a brief overview of the project structure. Check out our usage guide for everything the app can (and cannot) do: Not everything is as intuitive as it should be. Slicing a sheared 3D cube along the Y axis gives us cross sections that look like a shrinking square Until you see why the same thing happens in 3D: Slicing a sheared 4D cube along the W axis gives us cross sections that look like a shrinking cube It's not obvious why the cross sections of this sheared 4D cube look like a shrinking cube: You can always drop down one (or two) dimensions and inspect the same phenomenon. It's still hard to fully grasp high dimensional ideas like this, so another tool is dimensional analogy. The hope is that seeing objects in these two views simultanously aids understanding. Our goal was to let you view 4D objects through a projection view on the left (projected onto an intermediate 3D "screen" which is then projected onto your 2D screen) while seeing the cross sections on the right. (Right) The 3D cross-sections of the yellow hyper-plane with the 4D object (Left) Projection of a distorted 4D cube. Its original purpose was to teach an Introduction to 4D Geometry class for non-math majors. We developed this web-based viewer as our capstone project at St. Thinking about 4 spatial-dimensional geometry is fun, but it's very hard to develop an intuition for at first.
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