College of the Holy Cross
Department of Mathematics and Computer Science

• The Adventures of the Klein Bottle. It helps visualize the Klein Bottle and explains how it only exists in the fourth dimension. It's also pretty cool at the end how it brings you through the Klein Bottle to show what it's like to be inside and outside the bottle. --JT
  
• 3D to 4D shift--Carl Sagan. A video that discusses the 4th dimension. It tries to help us understand why we have so much trouble imagining a higher dimension with an example of 2D world trying to imagine a 3D world. Then the video talks about how we can picture 4D shapes in 3D using their projections.--PV
  
  
• How to Turn a Sphere Inside Out. This video helps to illustrate how one can take a sphere and manipulate it to turn it inside out in the fourth dimension. In addition, this process is completed without creasing or ripping the material used for the sphere.--MS
  
  
• The video is called Gluing a Torus and I thought it helped show both how a torus is a product of two circles and also how a torus is a rectangle with identified opposite sides.--AA
  
  
• 4D torus fly-through cross-section. This video shows 3D cross-sections of a 4D torus. It helps show that the cross-sections of a 4-dimensional object are 3-dimensional, which is useful for visualizing.--RM
  
  
• Advanced Mobius Strip Topology . - I thought I understood Mobius strips, but this guy adds a twist... actually two. Instead of one twist on a Mobius strip, he makes three and finds some interesting patterns including the fact they cross six times and they always have mirror images. It sometimes went a little over my head, but overall I found it interesting.--JY.
  
  
• Turning the Torus Inside Out. -The video I found is called "Turning the Torus Inside Out." It is a video of a torus being bent and twisted to turn the entire shape inside out. One "move" that the torus takes to do this is similar to the formation of a Klein bottle- where part of the shape is turned in to the inside of the other end. --SO.
  
  
• Coverings of the Circle. -This video demonstrates a covering of a topological space X is a topological space Y together with a continuous surjective map from X to Y that is locally bi-continuous. The infinite spiral is for example a covering of the circle. Every path on the circle can be lifted to the spiral. If a covering has a trivial fundamental group, i.e. it does not admit any non trivial closed paths it is called the universal covering. Here we see how a closed path on the circle is lifted to a non closed path on the spiral. Indeed the infinite spiral is the universal covering of the circle. --TC.
  
  
• Through one hole or two?. -The video shows an interesting transformation of the double torus to morph it on from one hole on a pole to two holes on the pole without breaking the knot to get it onto the pole. --RW.
  
  
• Punctured Torus. -This short movie shows a torus as a quotient of square surface and the punctured torus as a quotient of two bands glued together. --QH.
  
  
• The Alexander Horned Sphere. -The Alexander Horned Sphere, a sphere with curbed tubes that seem to branch out infinitely from each tube before it, is an interesting example of a topological space with its complexities reminding me somewhat of the Klein Bottle. The description mentions that it is a topological space homeomorphic to a sphere; however, the inside and the outside of this space are not homeomorphic to each other. --MM.
  
  
• Swan Homeomorphism. The video showed how two 3d images of swans and their mirror image are homeomorphic to unions of toruses. The smooth animated transformation exemplifies how a homeomorphism works visually. One of the swans at first did not look like it fit the double torus homeomorphism it was being transformed into, but upon my second watch it makes sense when the swan turns its neck and bill backwards into its feathers. --JB
  
  
• Moebius Transformations Revealed. This video is called "Moebius Transformations Revealed" and shows how Moebius transformations of a plane can be thought of as simple motions of a sphere.--KG
  
  
• Cyclist on Klein Bottle. This video essentially demonstrates what the Klein bottle is like in space, however the fact that the cyclist essentially has to go through the wall a few times shows how this cannot exist in the space we think in. Despite this flaw, it does give a good representation of what a Klein Bottle is and how we can imagine it. --MR
  
  
• No Magic At All: Mobius Strip. I had to try this experiment myself and the results are pretty interesting. This video shows the properties of a Mobius Strip (a band that has only one side and the boundary as a single circle) and contains four different experiments. We can make a Mobius Strip out of a long strip of paper by making a loop and flipping one end of the paper over with a half twist. For instance, first experiment shows that if we draw a line down the middle, this would get us back to the starting point but if you look at both sides of the strip, we would have drawn both sides of the paper. The second experiment shows that if you cut along the center of the strip, we would have created one large Mobius Strip instead of two strips. You won't believe it unless you try it but it actually works.--JY
  
  
• Braid Topological Math Puzzle. This clip shows a topological puzzle where the man braids a piece of plastic without cutting or using tape. --KS.
  
  
• Imagining the Tenth Dimension. An interesting and helpful way to describe and picture dimensions 1 through 10. Uses several examples from class including the Möbius strip.--SJ.
  
  
• Flowing Water Torus Knot. Its called Flowing Water Torus Knot. It shows the flow that water would take if traveling through a torus knot. The torus knot is sort of pretzel shaped and it is a continuous flow of water traveling through it.--RS.
  
  
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