This week, we discussed the topic of a surface's area being minimized. These
surfaces are called "Minimal Surfaces" and have the property that they generate
the smallest area locally on the surface. There are a couple of ways to describe
these surfaces, so I will use the general one that talks about curvature of the
surface, as well as the definition referencing neighborhoods of points.
Our Definitions:
Our first useful definition to describe these surfaces is:
A surface M $\subset \mathcal{R}^3$ is minimal if and only if (iff) its mean curvature is equal to zero at every point.
Now, you may be thinking to yourself, what does it mean for mean curvature? Well, the idea is similar to the topic of average value of a function from a calculus course. In particular, we take a surface which has been parameterized by a vector, $\overrightarrow{r(t)}$. Next, we want to know the unit tangent of the curve, call this $\overrightarrow{\mathcal{T}(t)}$. Finally, we describe the curvature as the relation $\frac{\overrightarrow{\mathcal{T}'(t)}}{\overrightarrow{r'(t)}}$, calling this $\mathcal{K}$. We want to describe this curvature in terms of $\theta$'s which is the angle with the normal vector and our plane. As discussed, we relate this to the average value of a function from your calculus class by finishing this definition by saying one checks the curvature by the following integral:
$\frac{1}{2\pi}\int_{0}^{2\pi} \mathcal{K}(\theta) d\theta$
Now, as I started by saying, there is another definition which is equivalent to the one just discussed. I see this one is a nice one for someone coming out of a calculus course, because if you have taking through multi-variable, this definition should feel more at home for you. Either way, I wanted to at least bring up this definition, since it goes more with our theme of straying away from calculus topics.
Definition Two:
A surface $C \subset \mathcal{R}^3$ is minimal iff every point $a \in C$ has a neighborhood, bounded by a simple closed curve, which has the least area among all surfaces sharing this boundary.
This neighborhood is where you have a set which contains a point. Around this point, you have a subset of your larger set, in which that smaller set is contained within. Basically, you try and take the smallest open set around some point. Lastly, this simple closed curve is a curve that starts and ends at the same point without crossing itself.
For my surface, I wanted to connect back to one of my favorite shapes, a pentagon. On the base, I created some pentagon. Above this pentagon, I created a 10-sided shape, a decagon. I wanted my surface to test how far you can change the base shapes for the bubble to form. By doubling the sides, I wanted to observe if this made any problem with the shape being created.
The surface should just be like my ruled surface shape, by the bases being a pentagon and a decagon. My object is less than three inches in length and width, with the height being a little over five and a half inches. I wanted to make sure that my shape will fit within the mason jar, so that is why the object has been constrained to these sizes.
Our Definitions:
Our first useful definition to describe these surfaces is:
A surface M $\subset \mathcal{R}^3$ is minimal if and only if (iff) its mean curvature is equal to zero at every point.
Now, you may be thinking to yourself, what does it mean for mean curvature? Well, the idea is similar to the topic of average value of a function from a calculus course. In particular, we take a surface which has been parameterized by a vector, $\overrightarrow{r(t)}$. Next, we want to know the unit tangent of the curve, call this $\overrightarrow{\mathcal{T}(t)}$. Finally, we describe the curvature as the relation $\frac{\overrightarrow{\mathcal{T}'(t)}}{\overrightarrow{r'(t)}}$, calling this $\mathcal{K}$. We want to describe this curvature in terms of $\theta$'s which is the angle with the normal vector and our plane. As discussed, we relate this to the average value of a function from your calculus class by finishing this definition by saying one checks the curvature by the following integral:
$\frac{1}{2\pi}\int_{0}^{2\pi} \mathcal{K}(\theta) d\theta$
Now, as I started by saying, there is another definition which is equivalent to the one just discussed. I see this one is a nice one for someone coming out of a calculus course, because if you have taking through multi-variable, this definition should feel more at home for you. Either way, I wanted to at least bring up this definition, since it goes more with our theme of straying away from calculus topics.
Definition Two:
A surface $C \subset \mathcal{R}^3$ is minimal iff every point $a \in C$ has a neighborhood, bounded by a simple closed curve, which has the least area among all surfaces sharing this boundary.
This neighborhood is where you have a set which contains a point. Around this point, you have a subset of your larger set, in which that smaller set is contained within. Basically, you try and take the smallest open set around some point. Lastly, this simple closed curve is a curve that starts and ends at the same point without crossing itself.
For my surface, I wanted to connect back to one of my favorite shapes, a pentagon. On the base, I created some pentagon. Above this pentagon, I created a 10-sided shape, a decagon. I wanted my surface to test how far you can change the base shapes for the bubble to form. By doubling the sides, I wanted to observe if this made any problem with the shape being created.
The surface should just be like my ruled surface shape, by the bases being a pentagon and a decagon. My object is less than three inches in length and width, with the height being a little over five and a half inches. I wanted to make sure that my shape will fit within the mason jar, so that is why the object has been constrained to these sizes.
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