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Thursday, July 14, 2016

Warren Buffett's Cube of Gold

In his 2011 annual letter to shareholders Mr. Warren Buffet, chairman of Berkshire Hathaway,  discussed the amazing rise in price of gold. Gold, at that time, was becoming extremely expensive (or valuable) and some shareholders wondered why Mr. Buffett did not buy gold to hold as the price increased.

Part of his response was:
"Today the world’s gold stock is about 170,000 metric tons. If all of this gold were melded together, it would form a cube of about 68 feet per side. (Picture it fitting comfortably within a baseball infield. [Bases are 90-feet apart.]) At $1,750 per ounce – gold’s price as I write this – its value would be $9.6 trillion. Call this cube pile A."
I remember reading that gold is extremely malleable and can be spread very thin. My thought was:

How much of our planet Earth could be covered by this hypothetical cube of gold?

To answer this question, I will use the data in the following table:

Factor or Formula
Density of gold
19.32 g/cm^3
Compute mass of hypothetical cube
Convert feet to meters
3.2808 feet per meter

Ounces to kilograms
35.274 ounces per kg
Convert mass to another unit
Ounce of gold to area
1-ounce covers 100 square feet
Radius of earth
6,353,000 meters
To find surface area of our planet
Surface area of a sphere (that is, Earth for our case)
 \(4 \pi r^2 \)
Formula (r is the radius of the sphere)

Step 1: How much gold is in this cube?

For a cube of 68-feet per side there are \( 68 \times 68 \times 68 \; = \; 314, 432 \; \; ft^3 \) of gold. From the table above we have the density of gold, but not in units of feet. Let’s first convert the density from grams to kilograms:
\[ Density = 19.32 \;\; g \; /  \; cm^3 \times \left( 100 \; cm \; / \; m \right)^3 \\  = 19,320 \; \; kg \;  /  \; m^3 \]
And let’s convert the density from meters to units of feet:
\[ Density = 19,320 \; \; kg/m^3 \times \left( 1\; meter\; / \; 3.2808 feet \right)^3 \\ = 547.10 \; kg \;  per \; ft^3  \]
Now we can compute the amount of gold in this hypothetical cube as: \( 314, 432 \times 547.10 = 172,025,747 \) kg of gold.

 Step 2: How big an area will the gold cube cover?

This site notes an ounce of gold ("about the size of quarter") can be pounded into a 100 square foot sheet, which is pretty thin if you think about it. There are 35.274 ounces per kilogram so our cube has
\[172,025,747 kg \times \left(35.274 \; ounces\; / \; kg \right) \\ =  6,068,036,199   \; \; ounces \]
and for that many ounces of gold, we have a surface area of
\[ 6,068,036,199 \; ounces \;  \times \left( 100 ft^2 \; / \; ounce \right) \\  = 606,803,619,900 \; ft^2 \]
That cube covers over 606 billion square feet.

Step 3: What is the surface area of Earth?

The surface area of earth is:
\[ S_{earth} = 4 \times \pi \times r_{earth}^2 \] and we use the radius of Earth given in the table to find:
\[ S_{earth} = 4 \times \pi \times \left( 6353000^2 \right) \\ = 507,186,370,915,240 \; m^2 \]
which we can convert to square feet as:
\[  507,186,370,915,240 \; m^2 \times (3.2808 \; feet \; / \;  meter)^2 \\ = 5,459,175,891,528,360 \; ft^2 \]

Step 4: What is the ratio of the gold cube area to the surface area of Earth?

As a fraction, the gold cube covers 0.0111 percent of Earth:
\[ \frac{606,803,619,900}{5,459,175,891,528,360} \times 100  = 0.0111 \; percent \]
 which is a pretty small relative to the size of Earth.

What’s the point?

While this cube costs $9, 600 ,000 ,000 ,000 it only covers a bit more that one-hundredth of one percent of the planet. Of course, the whole idea of gold as valuable is because it’s so rare, and these calculations confirm that. But it was fun to do the math, don't you think?

Check out:

Gold fun facts from the American Museum of Natural History.

Tuesday, April 05, 2016

The Dark Net by Jamie Bartlett

When the internet started, there were no homes connected to it. If you wanted to use email, and no one who didn’t already have email even knew what email was, you had to have an account on a computer at a research center like MIT. If you did, you could send email to others who had an account, you could transfer files from one computer to another (with FTP, file transfer protocol) and you could remotely log into another machine with telnet. And, there were news groups which were text-based lists of articles, called postings, of various topics such as food, humor, sex, etc. 

Fast forward thirty years to today and that limited research network is now a global network connecting users across the globe to one another. Anyone with a cell-phone can connect to anyone else. You don’t need an account on a school computer, your identity can be hidden (although not necessarily your internet protocol (IP) address) and you can access websites, forums, email, chats, etc. So, what’s life like on this new, global network?

The internet, of course, is filled with good things like news, email, social software, and, as you know, shopping on Amazon or eBay. But for Jamie Bartlett, the internet has a dark side filled with illegal activities and socially unacceptable sites packed with people who pay for the privilege to be part of a sex show. 

We begin this unseemly trip with Sara, a college student who thinks her computer offers some sort of anonymity, but it doesn’t. She puts on a striptease in front of her webcam on a public site and chats with viewers. She goes so far as to write her name on her naked body. Before she’s done entertaining (she’s not being paid) some of the viewers have “doxed” her: they find her real name, raid her Facebook account, send her nude photos to her friends and family, and essentially disrupt (maybe ruin) her life in the span of a just a few minutes. It’s a horrific description of an innocent girl’s demise written so graphically as to make you pity her naiveté and abhor the evil of these viewers who can’t resist shaming her. 

I was pleased to find usenet newsgroups are still around but then shocked to realize how misused they are on the dark net. The group makes its appearance where people openly discuss their desire to kill themselves. Suicide is a sickness and one might think, just maybe, this newsgroup would somehow help the mentally ill to avoid suicide and find help. No such luck. Mr. Bartlett introduces us to “Cami” who encourages Nadia, a 19-year old Canadian girl, to kill herself. Reading the dialogue one would think Cami is a kindred spirit and needs help, too. But we soon learn Cami is not a girl, not suicidal, and has no interest in helping Nadia. Rather, Cami is a middle-aged man, a nurse, husband, and father who has spent years encouraging distraught girls to kill themselves. He thinks five actually did. On the web, there are dangerous people.

If you want to purchase illegal goods like drugs or guns you’ll find them on the Silk Road. This is a site that only exists via the TOR router (The Onion Router, developed by the Navy) so users are almost impossible to trace and anonymity is all but ensured. The first Silk Road was run by Ross Ulbricht until he was arrested by the FBI on October 1, 2013. Ulbricht managed the site for a small fee amounting over time to $150 million in Bitcoins, the currency of choice for no traceability. There is now a Silk Road 2 showing that markets will spring to serve customers no matter what the merchandise. 

For the sex-side to this dark net we meet Vex who works on the Chaturbate site. Viewers can find sex shows with Vex and maybe a friend or two of hers will join in. She makes money from the shows and fans can tip her with Bitcoins. The site monitors who tips and how much so the bigger the tipper the more his (and most viewers are men) suggestions the models may perform. Some fans are quite fond of the various models and buy them gifts from the model’s Amazon wish list. 

The dark web exists, is easy to access, and contains things unavailable elsewhere. It’s a model of free enterprise, marketing, and business without regulation. For those interested in what happens when there are few rules, Bartlett has a written a fine guide to this side of the internet.

Monday, April 04, 2016

Book Review: A Sampler of Useful Computational Tools for Applied Geometry, Computer Graphics, and Image Processing By Daniel Cohen-Or (Editor), Chen Greif, Tao Ju, Niloy J. Mitra, Areil Shamir, Olga Sorkine-Hornung, and Hao (Richard) Zhang

Here is a useful book (the title is correct!) that covers a wealth of processing tools in a single volume accessible to experts and novices alike. Topics include analytic geometry, linear algebra, least squares solutions, principal component analysis and singular value decomposition, spectral transforms, solution of linear systems, and graphs and images, to name a few.  All topics are applicable to image processing and most are applicable to other areas as well.

Each chapter is stand-alone so the reader can learn the material quickly without the need to study earlier chapters to understand later chapters. If the reader doesn’t know what he needs, the table of contents shows the reader what’s in each chapter and whether the material might apply to his problem.

The text is lucid, easy to read, and follows a logical progression. Equations are explained and the authors provide ample figures and pictures to illustrate the concepts and discussions.

For example, the authors discuss how to determine is two lines intersect in 3-space with a step-by-step development of the mathematics and resulting algorithm. Next, they show how to fit a line to data points in a least squares sense and give pictorial examples in 2- and 3-space. The chapter on Least-Squares Solutions shows how to fit, say a curve to data points from an edge of an image and how to weight the data points to compensate for outliers. The chapter on spectral transforms provides an excellent discussion on image compression techniques such as the Discrete Cosine Transform and Laplacian smoothing. 

Incidentally, the book intuitively discusses how one can use Laplacian equations for image reconstruction and image manipulation. There is an image, for example, where most of the image is missing but with an image completeness algorithm based on smoothness (that is, find the best image that is smooth relative to the known data) one sees a fitted image quite close to the original. (Incidentally, it has always amazed me how much information one can discard from an image and yet still recover the original image, almost precisely.)

What is more, the book discusses Poisson panoramic image stitching and shows a beautiful example of city landscape before and after the processing. This tells part of the story of current digital cameras that allow one to take multiple pictures of a scene with each picture angularly offset from the other. The processed image is a smooth panoramic view that would be unobtainable otherwise. The stitching and smooth transition between images is both natural and unnoticeable.

On a slightly negative note, the chapter on topology is a bit weak.  Of course, topology is a complicated and involved topic and while the gist of it can be explained succinctly, applications require more space than the authors had to produce a well-balanced text. Still, as a reference that at least mentions this topic the authors do an admirable job summarizing the details.

In conclusion, this book is a good collection of algorithms and tools for computer graphics. The topics are useful, well-presented, and collected in a single book for easy access. If you work in this area, you’ll find this book beneficial.

(This review was originally posted to the Mathematical Association of America's Book Reviews and can be found here.)