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Lasagne with a Twist

Ruled surfaces have the property that any point on the surface is a part of a line in the 3D space that is also wholly contained in that surface. The constraint of having the entire surface made up of straight lines may make it seem like the only possible surface is a plane, but many curved surfaces can be made from straight lines. Some examples are cylinders, cones, and hyperboloids of one sheet. In fact, all three of these surfaces can be made by drawing lines between two circles where one is directly above the other. The type of surface obtained depends on the difference in degrees between the points on the circles that make up the lines on the surface. Helixes and double helixes are also ruled surfaces because they are composed of a straight line rotating around a point as it goes upwards. There are many other uniquely shaped ruled surfaces, but the one I created has to do with sine waves and pasta.
  The two functions I chose to make a ruled surface from were a sine wave going along the x axis and a sine wave going along the y axis. The specific parameterizations were \[x(t)=0, y(t)=t, z(t)=-10*\cos(20*t)\] \[x(t)=t, y(t)=0, z(t)=10*\cos(20*t)\] When shown as a surface, it looks like a lasagne sheet twisted 180 degrees in the middle.
When shown as a handful of lines making up the surface, you can see that the lines wobble back and forth before doing a complete 180 and then continuing to wobble.
My lasagne also has some interesting properties. Although it is not a doubly ruled surface (where every point has two lines on the surface that go through it), an additional line, \(y=x\), lies on the surface. This line was not created from drawing a line between the two curves, but it is still an extra line that is contained on the curved surface. Another interesting property is that as the lines in the surface get further from the middle, their slope approaches zero because the sine waves get further and further apart, but still have the same difference in height. Also, if you were to graph the slope of the lines, the graph would go back and forth between positive and negative with its amplitude getting bigger as it approaches zero, and then the slope suddenly has an infinite discontinuity at zero before returning to going back and forth between positive and negative.
  The reason I chose these curves was because the surface made from them reminded me of my favorite fictional cat's meal of choice. Garfield the cat loves lasagna. I love Garfield. Therefore, by the transitive property, I love lasagna. I don't know how to make an entire dish of lasagna from straight lines though, so I decided to go with a lasagne sheet instead. A simple lasagne sheet was too ordinary, though. I wanted it to have some sort of twist. So I gave it one. The resulting surface was lasagne with a twist. I'm sure that if Jon Arbuckle served Garfield lasagna made with my twisted lasagne sheets, that orange cat would be amazed at the mathematical properites of the food he was eating.

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