MTH 495 – Blog Post 6 – Spider Geometry?!

The following problem comes from the NRICH website at http://nrich.maths.org/2365/2365. Here is the problem:

The Spider and the Fly

Stage: 4

A spider is sitting in the middle of one of the smallest walls in my living room and a fly is resting by the side of the window on the opposite wall, 1.5m above the ground and 0.5m from the adjacent wall.

The room is 5m long, 4m wide and 2.5m high.

What is the shortest distance the spider would have to crawl to catch the fly?

If the fly walks down the wall, is there a point at which the spider would be better changing its route?

I am attaching my work below. However, to start I was fairly certain that the spider was going to have to take an angled route to the fly over then entire distance traveled. This way, the spider would be covering the horizontal and vertical distances at the same time. Even so, I wanted to look at a few different cases to see if I might be missing something.

Here are the cases I came up with, where the vertical distance was covered diagonally over various portions of the horizontal distance:

1. 1^st wall
2. 1^st & 2^nd walls
3. 2^nd & 3^rd walls
4. 1^st & 3^rd walls
5. 1^st, 2^nd, & 3^rd walls
6. 2^nd wall
7. 3^rd wall
8. Horizontal distance 1^st, then vertical

Okay, as I was saying, I was fairly certain that the 5^th case listed was the best option. Also, I figured that there were other possible cases that needed to be considered but I figured that this was at least a starting point.

To figure out the distances, I essentially used the net containing the three walls in question, unfolded it, and use the Pythagorean Theorem to calculate the diagonal distances.

I started with case 8 (Worst case scenario! So I had something to compare the shorter distances to), and I found that the distances for cases 8, 5, 7, 3, 2, & 1, in the order I worked on them, was:

8 – 7.75m

5 – 7.50416551m (This is almost 0.25m shorter than the worst case.)

7 – 7.559m

3 – 7.505678886m

2 – 7.504462863m

1 – 7.515564437m

I stopped looking at cases here because I was fairly certain that I had found the shortest distance. Then I thought of another case. What if the spider had a thread of silk that went direct from where it was standing, to the fly’s location? This was tricky, and I am still not 100% certain I got the right distance. I don’t like that I can’t see it clearly in my head, but here is how I see it.

First, imagine there is a string that goes directly from the wall the spider is on, straight across, as in parallel to the floor, to the wall the fly is on. That distance would be 5m but the spider would still have to be travel 0.5m horizontally and 0.25m vertically, a total of 5.75m. Well, this is already much shorter, but could it be shorter if the thread was attached to the fly?

To get this distance, I used the Pythagorean Theorem to get the diagonal distance on the fly wall from the point straight across from the spider to the location of the fly. This gave me a distance of 0.5590169944m. Now, if you picture our right triangle as being slightly rotated, so that the base is still the line from the spider's position to the point on the fly’s wall, directly across from the spider, then the side of the triangle that gives us the height would be that diagonal lime we just calculated. The hypotenuse of this triangle then, would be our silk thread. To get the distance of this line we have:

Sqrt((0.5590169944)² + 5²) = 5.031152949m

Now, I maintain that this is the shortest possible distance from the spider’s current location, to the fly’s current location, without creating a worm hole or something like this.

However, there are other cases, that are far more likely, that could be shorter than the one using the three walls. These are using the ceiling or using the floor. Since the fly is closer to the ceiling than the floor, and the spider could get squashed on the floor, and the ceiling is far less likely to have huge obstacles on it, like a couch for example, I decided to go with the ceiling.

This gives us:

Sqrt((1.5)² + (7.25²) = 7.403546447m

Which is the shortest distance of the more likely cases/scenarios. Not only that, for the reasons already discussed and many more, it is also likely the safest path for the spider to take. Even so, I personally like the string theory the best.

Hmmm . . . I wonder if it was a spider we, instead of just a straight, lone, and magically gravity resistant string, if we could use graph theory to find the shortest distance, assuming the spider had to walk along the treads and not along the top of the web, and what that might do to our distance. Would the spider give up on the web and just take the wall after all?

Hey, to get a better idea of my approach, check out the notes below. If you see something I missed or anything else worth talking about, let me know.

Thank you for reading,

Jerry

MTH 495 – Blog Post 5 – Algebraic Rigor!

In one of the books that I have read recently, either “The Calculus Gallery: Masterpieces from Newton to Labesque” by William Dunham or “Isaac Newton” by James Gleick, it was said that Isaac Newton was known for his rigor with algebra. Looking up the definition of rigor on www.oxforddictionaries.com/us/, I found “The quality of being extremely thorough, exhaustive, or accurate.”

This, in addition to several experiences in class recently has piqued my interest in just what it means to have algebraic rigor. With this in mind, or discussion in class on Tuesday was on Georg Riemann and how he worked to define distance as a function and came to the conclusion that any distance function must have 3 properties:

1. D(x,y) = D(y,x)
2. D(x,x) = 0
3. D(x,y) + D(y,z) >= D(x,z)

We were then asked what function is normally used to determine the distance between two points on the coordinate plane. As is:

For point A(x₁,y₁) and point B(x₂, y₂), the distance, D, is:

D(A, B) = Sqrt((x₂ – x₁)² + (y₂ – y₁)²)

Okay, well how hard could it really be to prove that the distance formula fulfills the third property? Turns out, this is one problem that requires some significant algebraic rigor! After hours of work, I still haven’t finished and had to set it aside to work on this post. I am not ready to throw in the towel, I just needed to take a break. I am going to attach my work below, so take a look and let me know if you see anything I am missing or if you see a way to reduce the number of terms I am dealing with. I did use substitution to allow me to continue to work through the problem without having to rewrite everything over and over again. However, it seems that I have now reached a point where nothing is going to cancel out unless I expand everything out. I wonder if there are tricks for this that I am missing.

Anyways, take a look and let me know what you think.

Thank you for reading,

Jerry

MTH 495 – Book Review - Blog Post 4

The Calculus Gallery: Masterpieces from Newton to Lebesque 

So the book I chose to read was, “The Calculus Gallery: Masterpieces from Newton to Lebesque” by William Dunham. I chose this book because of the focus on Calculus. Since I am going to be taking Calculus 3 next, I wanted a book that would be interesting but also give me some refresher prompts on Calculus as well as teaching more on the subject. I have to say, on all of these points, this book really delivered!

This book was the perfect companion to this course because it mimicked the structure so well. It started at the beginning of the subject, or as best as we can tell, and traveled through time, into the future, making stops along the way to highlight key figures and some of their most significant discovers. As such, the book covers most, but not all of the mathematicians who have made an impact on the subject of Calculus. As the author points out, like most galleries, this one would not be able to contain every individual that had made a contribution.

As you might expect, the book starts with Newton and Leibniz and the controversy over who was the first and then traveled forward in time from there. One point that I found particularly interesting was that William Dunham did a great job of pointing out the differences in approach between Newton and Leibniz and the value of each and eventually, the weaknesses.

Did I just say that there are weaknesses in the theories that were presented by the founders of Calculus? Yes, I did, and this is another fascinating part of this book. The author makes a point to inform the reader that the early foundations of Calculus were built upon instinct and intuition and that much of the work of those who came after Newton and Leibniz was directed towards removing these vague concepts that just seemed to work and replacing them with solid theorems and proofs.

Of course, the book contains sections on such greats as the Bernoulli brothers, Euler, Cauchy, Riemann, and so many others. However, I think that my favorite was Karl Weierstrass. Not only did Weierstrass go from being an unknown math professor from an unknown town to become a renowned mathematician at the University of Berlin, but he did so with style and an intense focus on precision in proofs that we key to removing the last bits of unproven intuition that was left. Oh, and did I mention that he was because he suffered from severe vertigo, he was able to lecture from a chair while a chosen student wrote his words on the board!

In addition to all of the great history and discussion of the figures in it, the books covers in detail the key mathematical contributions they made by showing the math, the proofs, and the papers written by these great mathematicians. Mr. Dunham also goes so far as to discuss how these ideas have changed over the years and how they are now taught in a modern day Calculus class.

It is this part of the book that can make it a challenging read. While I am familiar with the subject, it has been a sufficient period of time since my class where I was not able to follow all of the points. I think, for this reason, the book would be a better read after completing Calculus 1 & 2, than it is as a refresher before going into Calculus 3.

In the beginning, I read each section on the math, doing my best to understand the first concept before moving on to the next. However, at the end, this became more and more challenging from the perspective of my ability to focus on what each portion of the proof was saying. Likely, this is for good reason, as the math being discussed and the discoveries made, become more and more complicated and at the same time, more and more focused on certain aspects of Calculus.

Overall, impression? A great book for any math major, especially after taking Calculus 1 & 2. However, if you are looking a book that are a lighter read on the origins of Calculus, I highly recommend, “Isaac Newton” by James Gleick. Also, if you are interested in a silly book that shows how Calculus can help you survive the zombie apocalypse, I recommend “Zombies and Calculus” by Colin Adams.

Thank you for reading,

Jerry