3D Basketball

I think its been around before this, but I noticed during this year's NBA Finals the use of some kind of 3D technology. It's very impressive. I didn't know how they did that and I still don't, but I have a few leads.

This is possibly not the kind of math application you can share all the details with a class, as its seems a little complicated. In fact, for the NBA to pull this off takes takes several cameras and expensive software.

I first came across some information on how this is done while reading the June 30th issue of Sport Illustrated, "Computers break down the images into voxels (3-D pixels) and the view from any point on the court or field can be re-created into a three-dimensional, 360-degree video of the action."

That didn't tell me a lot, but it did give me the term "voxels" that I could Google.

From WhatIs.com (http://whatis.techtarget.com/definition/voxel) I found this.

"A voxel is a unit of graphic information that defines a point in three-dimensional space. Since a pixel (picture element) defines a point in two dimensional space with its x and y coordinates, a third z coordinate is needed. In 3-D space, each of the coordinates is defined in terms of its position, color, and density. Think of a cube where any point on an outer side is expressed with an x , y coordinate and the third, z coordinate defines a location into the cube from that side, its density, and its color. With this information and 3-D rendering software, a two-dimensional view from various angles of an image can be obtained and viewed at your computer.

Medical practitioners and researchers are now using images defined by voxels and 3-D software to view X-rays, cathode tube scans, and magnetic resonance imaging (MRI) scans from different angles, effectively to see the inside of the body from outside. Geologists can create 3-D views of earth profiles based on sound echoes. Engineers can view complex machinery and material structures to look for weaknesses."

So getting to see NBA replays in 3D is cool, but probably a little trivial. It was interesting to see the application to medicine and geology.

There turned out to be quite a few hits on "voxels". A lot of them dealing with buying the above mentioned software.

Regarding medical applications, the Scientific American website had an interesting article (http://blogs.scientificamerican.com/observations/whats-a-voxel-and-what-can-it-tell-us-a-primer-on-fmri/) on how voxels can be an improvement over regular MRIs.

From another site I found that while "pixel" is short for picture element, "voxel" is short for volume element. That makes sense. Much of the other information I found does not make a lot of sense to me.

From Webopedia.com I found this info: Voxelization is the process of adding depth to an image using a set of cross-sectional images known as a volumetric dataset. These cross-sectional images (or slices) are made up of pixels. The space between any two pixels in one slice is referred to as interpixel distance, which represents a real-world distance.

This article quickly spiraled away from my level of understanding, but this part kind of made sense. For the NBA 3D replay, perhaps each "slice" is the 2D view from a particular camera. Combining this with several other cameras gives various slices. The distances between those slices constitute the third dimension for various points. I don't know, but I might be onto something. 

Next week will be a new topic, or possibly, if I gain any big insights into voxels, maybe Part 2 of this topic.

My Book

I might have mentioned before that I was writing a book. It is meant for high school math teachers. The best way to describe it that it might be the reply to students that ask, "Where are we ever going to use this?"

It isn't really out yet. It's supposedly being released in the fall. It is called "Math for Real Life" and is published by McFarland Publishing. The company has been great and know what their doing. It is been an interesting education for me as to how publishing works. In the contract they made clear what they would do and what I would do. Also, they let me know what things I might have input on, but that they would be making the final decisions.

For example, I wanted a little different title. They apparently didn't care for my title. I also could have input on the cover, I didn't really have any ideas on that, and they came up with something cooler than I would have ever come up with.

After I signed a contract with them I sent them my manuscript. They said there would be several months in which I wouldn't hear much from them as they prepared the book and not to get anxious and bug them. It was a little disconcerting, but I remained patient. Supposedly I'll get it back in a bit and make any final corrections and then I'm required to come up with an index.

Anyway, they've been very professional and great to work with. Its been a fun process. When its actually available, which I think will be in a couple months, I'll mention that in the blog.

Wedding Pictures

I was at a wedding this weekend. Counting parents, bridesmaids, etc. there were about 15 people during a photography session right before the ceremony. It seemed to go on and on with various combinations. I of course thought, "I wonder how many combinations there are if we do all the possibilities."

After a little thought, I figured it would be:

C(15,0) + C(15,1) + C(15,2) + C(15,3) + ... + C(15,15)

Granted, some of these would be unlikely, e.g., C(15,0), but this expression would at least figure the upper bound. Then I was told that the total could be found with 2 to the 15th power. I never knew that. If true, that is cool. I don't know if that is true, because I have not worked out the proof. I assume I would not be smart enough to do so.

I did try out a few cases to see if it worked for them.

c(2,0) + c(2,1) + c(2,2) = 1 + 2 + 1 = 4. This is equal to 2^2.

Also c(3,0) + c(3,1) + c(3,2) + c(3,3) = 1 + 3 + 3 + 1 = 8. This also happens to be 2^3.

It's looking pretty good. As I look at this, I'm seeing Pascal's Triangle. So there is another thing I wasn't previously aware of. I'm learning.

Anyway, most combinations of ways to group those 15 people was 2^15 = 32,768.

More Water Towers

Last week we took a look at how water towers work. Certainly students would want to know something about them before they would feel very motivated to do any math with them. Since it took me literally decades before I even had the slightest idea how they worked, I'm assuming most students don't know much about. At least I'm hoping that. 

I used to think maybe they were open at the top and caught rain water and stored it. But then there would be birds and stuff getting in there. And we're drinking that? Luckily how I thought they worked isn't at all how they work. You can read last weeks blog for some basic info on how they do in fact work. 

Here is a great example of a math application. It is a combination cone and cylinder. I tried to blow it up so you can see the numbers. Of course you can simply make up problems with numbers, but its nice to have numbers of an actual thing - even if the thing is just a picture from the internet.

Our town has a cylindrical water tower whose base sits on the ground. A good project would be to estimate the number of gallons it would hold. You could estimate the the diameter by first pacing off the circumference and then doing a little math. Then you could just estimate the height by eyeballing it, or better yet, doing some trigonometry. Granted, it would be a pretty rough estimate, but a nice project. I'm sure the water department, or someone, has the actual numbers. You could then get those and compare your estimate to what they say. 

I saw a company on-line that said they had towers, "Available in diameters from 11 feet (3.3 m) to 204 feet (62.2 m) and capacity from 20,000 gallons (75 cu m) to over 6 million gallons (22,700 cu m)". They are the self-proclaimed "premium water and liquid storage technology leader", so they must know their stuff.

This is a picture from their website. Oddly they don't mention the height, So a question might be:  For a given diameter, say 50 feet, what height would be necessary to have a 100,000 gallon tank? I looked up the fact that there are 7.48 cubic feet in a gallon. There you go. A great application.

A look on line shows that there are certain fairly standard shapes, but quite a few atypical designs, lending themselves to using several different volume formulas.

Water Towers

Do you know how water towers work? I have never known. I asked some people recently and they didn't know either. They're filled with rain water? It looks like a lot of water, but not nearly enough to last a city for very long. What happens if/when they run out? So many questions.

Here is a crash course that isn't going to cover everything, but it may answer a few questions you might have.

I thought the process kind of starts at the water tower, but it doesn't. The water we drink starts off at a water treatment plant. In a college biology class, we had a field trip to one. It was my only college field trip ever. Since I have all these questions now, I must not have paid very good attention.

Depending on the need, water is then pumped to the city or up into the tower. The water in the tower is just there in case it is needed. It holds a lot, about 50 swimming pools full, but still wouldn't last a city very long. It is probably enough for about a day. It can be used if power goes out, which would shut down the pumps. It can also be used for emergencies, such as fighting a fire. It is also used as a supplement if what is being pumped from the water treatment plant isn't enough.

As mentioned, it can be used in the case of a power outage. That is because water comes out from the tower by the force of gravity. That is why they are always located on the top of a hill. The water inside can only flow to points at a lower elevation than the tower itself.

There are plenty of good math applications involved here. Obviously, finding the capacity of the tower can be important. As can be seen, they might be conical, spherical, or cylindrical, or even combinations of shapes.

Finding the amount of pressure of the water flowing from the tower would be important. Every foot in height supplies 0.43 pounds per square inch. Thus, the higher it is, the more pressure. We'll look at these issues a little more next time.

Until now, I obviously never took the time to look into this topic. I got a lot of my information from several sources, but http://people.howstuffworks.com/water.htm was probably the best. That is where the diagram below is from.

A = From Treatment Plant
B = Pump
C = Water Tower
D = To Customers

The Bermuda Triangle

The Bermuda Triangle. It is the site of many shipwrecks and plane disappearances. Many dispute whether this is really all that extraordinary. After all, as can be seen, it encompasses a large area. Plus it is an area that has a lot of hurricanes and other disturbances in the weather. So all the mishaps could be something weird going on or maybe not that strange at all.

As mentioned, it is a large area. How large in fact. We have the standard A = 0.5bh formula, but it isn't easily used for this. So, this is a great application of Heron's formula.

To review, for any triangle with sides a, b, and c; what is called its semiperimeter is s = (a+b+c)/2.

The area then is A = square root of (s(s-a)(s-b)(s-c))

So, lets try it out. The value of s turns out to be 1,511. Plugging the numbers for a, b, c, and s into Heron's Formula gives a value of 437,600 square miles.

Different people give differing locations for the sides and vertices of the triangle, so there will be some differences. Also, we are not taking into account the curvature of the earth. Even so, it's a nice application.

Pronouncing Mathematician Names

Last time I took a shot at ranking the top mathematicians of all-time. One thing that has always bothered me is not knowing how to pronounce mathematicians names. Some are tricky. You might not know how to pronounce Euler or Euclid, but at least you figure they would be roughly the same. They aren't.

These names often don't get said aloud. If a person comes across a mathematician's name, its probably because of reading it. As a service, I decided to look up pronunciations of some of the trickier ones and list them here. If a math teach is going to talk about famous mathematicians, and I think we should, we at least owe it to them to get their names right.

I'm going with dictionary.com for these. If you think any of these incorrect, you'll need to take it up with them.

In no particular order, here we go:

Euler: OI-ler

Euclid: YOO-klid

Gauss:  gous

Descartes:  dey-KAHRT

Lagrange:  la-GRAHNZH

Leibniz:  LAHYB-nitz

Poincare:  pwan-ka-REY

Riemann:  RE-mahn

Pascal:  pa-SKAL or pa-SKAHL

Fermat:  fer-MAH

Galois:  gal-WAH

Bernoulli:  ber-NOO-lee

Cauchy:  koh-SHEE

Hermite:  her-MEET

Laplace:  la-PLAS

Fourier:  FOO-ree-ey

Huygens:  HAHY-guhnz