Redo: Minimal Surface

I chose to redo the minimal surface project due to the last one being a little bit too complicated to print, and implement. The Schwarz P-Surface was very cool and interesting but it was nearly impossible to print and it resulted in many hours spent trying to print it or assemble it in different ways. Now we are working with two simple shapes, a circle and a square. The improvements here are the simplicity and the ability to actually form a minimal surface between the shapes I am using to demonstrate the concept. It is very hard for me to use Openscad and it is also very hard to tell what will work on a printer and what won't, sometimes it seems like it simply depends on the day! Jokes aside I think this will demonstrate the concept better and will be easier to work with making it a better project in terms of presentation and utilization.

The utilization of simple shapes, easy to understand curves, and the like are key to having a good minimal surface project in my opinion. So, reducing the complexity and making it easier to see also aids in the understanding of whats being presented as well. I also think that it will be interesting to see the shape that forms between a smooth surfaced shape with rounded edges being the circle and the square which has more sharp edges. All in all I think the concept of the Schwarz P-Surface is very interesting but when it comes to being able to portray that in a 3-D print, it falls apart. Literally my surface fell apart many times and using glue was very strenuous. I also like this rendition because it seems like something you could actually see in a toy store or kids playing with, compared to the Schwarz P-surface which ended up looking like some sort of sci-fi orb that almost no one recognized.

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Do Over: Integration Over a Region in a Plane

Throughout the semester we have covered a variety of topics and how their mathematical orientation applies to real world scenarios. One topic we discussed, and I would like to revisit, is integration over a region in a plane which involves calculating a double integral. Integrating functions of two variables allows us to calculate the volume under the function in a 3D space. You can see a more in depth description and my previous example in my blog post, https://ukyma391.blogspot.com/2021/09/integration-for-over-regions-in-plane_27.html . I want to revisit this topic because in my previous attempt my volume calculations were incorrect, and my print lacked structural stability. I believed this print and calculation was the topic I could most improve on and wanted to give it another chance. What needed Improvement? The function used previously was f(x) = cos(xy) bounded on [-3,3] x [-1,3]. After solving for the estimated and actual volume, it was difficult to represent in a print...

Minimal Surfaces

Minimum surfaces can be described in many equivalent ways. Today, we are going to focus on minimum surfaces by defining it using curvature. A surface is a minimum surface if and only if the mean curvature at every point is zero. This means that every point on the surface is a saddle point with equal and opposite curvature allowing the smallest surface area possible to form. Curvature helps define a minimal surface by looking at the normal vector. For a surface in R 3 , there is a tangent plane at each point. At each point in the surface, there is a normal vector perpendicular to the tangent plane. Then, we can intersect any plane that contains the normal vector with the surface to get a curve. Therefore, the mean curvature of a surface is defined by the following equation. Where theta is an angle from a starting plane that contains the normal vector. For this week’s project, we will be demonstrating minimum surfaces with a frame and soap bubbles! How It Works Minimum surfac...

Ruled Surfaces : Trefoil

Ruled Surfaces : Trefoil A ruled surface is a surface that consists straight lines, called rulings, which lie upon the surface. These surfaces are formed of a set of points that are "swept" by a straight line. This is relatively intuitive once you see a good visual, but can be a bit abstract without that concrete example. A very basic example of a ruled surface is a cylinder; if we have a straight line and move it in a circle we create a cylinder made entirely of straight line. Note that the surface will only be a cylinder if all the lines are parallel. If the lines are not parallel we can create hyperboloids and cones depending on how much we have rotated. The rotation we are describing here is not a simple turning action, but more of a twisting motion—less like rotating a can by turning it and more like wringing out a washcloth by twisting it. Specifically, a cylinder is essentially two circles connected by rulings, if we keep one of the circles...

MA 391 Composition and Communication

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