Aquatic Humans, Gorillas, and Crocs

There are interesting arguments and evidence supporting the theory that an ancestor of homo sapiens had a semi-aquatic existence at one point, where they hung out in the water a lot, at least part of the time. If you haven't heard about this yet, here's the wikipedia entry, and here's a nice 17 minute TED video with Elaine Morgan.

One counter argument is that in Africa, where humanity evolved, semi-aquatic living would a have a risk of being attacked by aquatic animals such as crocodiles. I wondered what the distribution of gorillas and the distribution of crocodiles in Africa looked like. With a couple minutes of googling, I found two maps purporting to be the distribution of Gorillas in Africa (src), and the distribution of Crocodiles in Africa (src, the only one I could find). I coarsely scaled and overlapped them, and this is the result:

In this image, the crocodile distribution (grey) is widespread, and the Gorillas (red) appear to not share the same area as Crocs in the West, but they do in the East. There are two species of Eastern Gorilla, the Eastern Lowland Gorilla, and the Mountain Gorilla. The Mountain Gorilla lives nowhere near Crocs, and the Eastern Lowland Gorilla lives in lowland and mountain rainforests and subalpine forests.

The Western Gorillas do not generally seem to overlap with crocodiles and their habitat, and recently the Wildlife Conservation Society found more than 100,000 previously unreported gorillas have been living in the swamp forests of Lake Tele Community Reserve.

I don't know why I thought to look at gorillas first, as Chimpanzees are an even closer relative of humans, so I did the same thing with an image (src) of their distribution in Africa. The following is the same image as above, with the chimpanzee distribution in green:

As you can see, almost the entire distribution of Chimpanzees does not overlap with Crocodiles!

These images were taken from arbitrary sources on the internet, are coarsely scaled against each other, and I know nothing of their accuracies in distribution.

If these images are a coarse approximation of reality, it appears that the nearest neighbor species of Ape spends their time away from crocdiles. These maps in of themselves prove nothing, as these are maps of modern species and of questionable accuracy. However, if our human ancestors were living in similar regions as present day Apes, these maps suggest it is reasonable to suggest crocodiles weren't a problem for semi-aquatic ancestors of our genus, at least, not enough of one.

Burton MacKenZie www.burtonmackenzie.com

Do Customs Agents have the right to your DATA?

I am not a lawyer. Seek local legal counsel first; you are responsible for your actions, not me.

As people become increasingly technological we proportionally keep more personal records and information on our portable data devices (e.g., laptops, iPhones, PDAs, etc).  Previous to the data revolution, our deeply personal information (e.g., love letters from your wife, your financial dealings, etc) was safe at home, at a bank, or another physically disparate location. Customs did not have jurisdiction to look through information/data other than what you brought with you, and what their existing records on you were.

We are now in the position that some of your data is not necessarily on a portable device, but it is accessible by it.  For instance, your google-drive might contain private information about you, and you can access it via your iTouch.  A customs agent should have no more ability to search your remotely-accessible files any more than they could send somebody over to your place to retreive some personal files they would like to see.  You may have your portable configured to link to your gdrive after a password is entered, but you should not provide the password to it for a search any more than you should provide a key for the customs agent to go search your home.

Be prepared - you may get turned back at the border - but they will be unable to take a copy of your filesystem because you are not carrying your filesystem; it is not on your person.   Even if your portable were seized and confiscated (legally or illegally), they still would not get the remote data, and that is how it should be.   In my country we have legal rights respecting protection from unreasonable search and seizure.  Your remote files are not within the jurisdiction of a border patrol; they cannot search them.

Nobody likes a hassle, but if people do not stand up for their rights they will ultimately lose them, tiny step by tiny step.  Stand up for your rights; refuse illegal searches.  Keep your data remote.

If any lawyers for various nationalities want to comment and tell me what their laws/jurisprudence are, I'd be happy to hear from you.

Burton MacKenZie www.burtonmackenzie.com

Sums of Squares: Being Complex makes it Easy!

Did you know that (42² + 111²)(2² + 5²) = (471² + 432²) = (639² + 12²)?

Neither did I until just a minute ago.  It turns out that the product of two sums of two integer squares is also equal to two sets of sums of other integer squares, and it's easy to figure out what they are.

Leonardo Pisano Fibonacci showed this in Proposition 6 of "The Book of Squares" in the year 1225, and I have read that even Diophantus was aware of it some 1800 years ago, but it seems it was Leibnitz who first applied complex numbers to this problem.  Complex numbers make this problem simple.

Take the general case,

(a² + b²)*(c² + d²)

which factors to the complex roots

= [(a + ib)(a - ib)]*[(c + id)(c - id)]

rearrange the factors to get

= [(a + ib)(c + id)]*[(a - ib)(c - id)]

which multiplies out to

= [ (ac - bd) + i(ad + bc) ] * [ (ac - bd) - i(ad + bc) ]

but since this is a complex conjugate, it simplifies to

= (ac - bd)² + (ad + bc)²

That is, given any product of sum of squares, you can easily find what other sum of squares represents the answer.

Further, we can also re-order the factors to give

= [(a + ib)(c - id)]*[(a - ib)(c + id)] 

= [ (ac + bd) + i(bc - ad) ] * [ (ac + bd) - i(bc - ad) ]

= (ac + bd)² + (bc - ad)²

Which is yet another solution represented as a sum of squares!
Try it!  (42² + 111²)(2² + 5²) = (471² + 432²) = (639² + 12²)

Burton MacKenZie www.burtonmackenzie.com

Happy Pi Day!

In honour of Pi Day, I ate a whole Pi!

Burton MacKenZie www.burtonmackenzie.com

Pi Day Update: I am disturbed by this video. I love it!

The Sun as seen from other Planets

If you were standing on another planet in our Solar System, the size of the Sun's disk in the sky would be different than the size it looks from Earth.  Below is an image I made of how large the Sun looks from every planet.  Earth is shown in a brighter yellow to help your sense of scale.

From the perspective of lonely Neptune, our Sun just looks like a bright star.   Pluto is not included in the diagram, but its visage of the Sun is the same as Neptune's: a bright star in the distance.

I'm already freaked out by how big the Sun is as seen from Earth - imagine being on Mercury, confronted by that giant ball of nuclear radiation!  Its visible disk is almost 7 times bigger than it is on Earth!

Burton MacKenZie www.burtonmackenzie.com

p.s. Here is an earlier post about Mercury.

Why do so many people wonder if Ann Coulter is a man?

Last year I was playing with the automated Gender Guesser.  There were stirrings on the net that Ann Coulter, an american who makes a living by spewing controversial nonsense and hatred to the ignorant masses, was really a man.  I went to her website and input some of her posts into the Gender Guesser.

The Gender Guesser is not a predictor.  It cannot predict definitively whether or not somebody is a man or a woman any more than being a smoker predicts you will get lung cancer.  At best, it might give results like "In a room of 100 people, if they all wrote this way, 70 (-ish) of them would be men".  That is, there is a lot of room for error, nor can it predict an individual.

Nonetheless, as a lark I published its claim (with appropriate disclaimers of accuracy) that her own writings betrayed her as a man because of what the Gender Guesser reported.

Sadly, outside of short spikes of popularity, this has become my most popular post on my entire blog, a popularity almost exclusively based on daily google searches for "Ann Coulter is a Man" or "Is Ann Coulter a Man?".  I am not proud of this - my blog is mostly math and science based.  It is a thorn in my side that my most popular post is due to people searching for information on her true gender and that, on average, it far outcompetes my regular posts.  I considered removing the original post entirely, but that didn't seem right, either.  I was reluctant to even add this post, but in the end decided to publish the bizarre website traffic happenings that I've been seeing for months.  This has been going on for a while but I didn't want to lead in January by mentioning it.  Mentioning Ann is a bad Juju way to start a year.

Ann, I don't care if you're really a man or not.  If you want to say you're a woman that's fine with me.  Saying you were a man on the Lou Dobbs show didn't help, even if you were joking.  I really do wonder, though, why so many people every freakin day come to my blog from google searches wondering if you're a man.  WTF?  No, seriously.

Burton MacKenZie www.burtonmackenzie.com

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Peasant Finger Multiplication Proof for all Bilaterally Symmetric Species

There is a technique for multiplying numbers between 6 and 10 on your fingers called peasant multiplication (graphic illustration).  I explained how to do it two years ago, and I mentioned an easy proof of it.  Since I was subsequently asked for the proof, I present it now with an improvement.  (If you don't already know how to perform the multiplication, please check the links above, first)

Why Peasant Finger Multiplication works for any Bilaterally Symmetric Species

As long as your fingers (or tentacles) on each hand (or appendage) are bilaterally symmetric and you count in a base defined by your total number of fingers, this method will work.  First, let's start with definitions:

Since "digits" has different meanings in physiology and math, I will say "digits" when I mean mathematical place-value digits, and "fingers" when I mean dangly meat appendages with which you are counting/multiplying.

let n = factor #1 to be multiplied

m = factor #2 to be multiplied

p = product n * m

d = the number of fingers on each hand (total fingers on both hands = 2*d)

Since we (humans) count each number (n or m) from 6 to 10 on a hand with 5 fingers, the actual number of fingers used on a hand would be n-5, or more generically for any species,

a = n - d    (number of fingers used on one hand)

b = m - d   (number of fingers used on other hand)

If the product p for which we are searching is defined as

p = n * m

then

p = n * m = (a + d)*(b + d) = ab + d*(a+b) + d²

With peasant math, the tens digit (base 10) is represented by adding the number of fingers on each hand representing digits.  Here I multiply it by 10 to represent that it will be placed in the 10s digit:

p₁₀  = 10*(a + b)

or more generally for any species, the "d's" digit is,

p_d  = 2*d*(a + b)

Similarly, for the value of the ones digit, you add the remaining (unused) fingers on each hand:

p_ones  = (d - a)*(d - b)

which expands to

p_ones  = d² - d*(a + b) + ab

In peasant math, the product p is acheived by combining the 10s digit with the ones digit:

p = p_d + p_ones = 2*d*(a + b) + d² - d*(a + b) + ab

= ab + d*(a+b) + d^{2 =}  (a + d)*(b + d)

= n * m

which is the original definition of the product you are seeking to find.  

Peasant math works, regardless of your species, as long as you're bilaterally symmetric and your numeric base is that of your total number of fingers, tentacles, or tough scaly claws. 

Burton MacKenZie www.burtonmackenzie.com

Multiplying big numbers using Cosines

I came across a technique for multiplying big numbers recently, called prosthaphaeresis [1], that predates the use of logarithms.  It relied on the use of pre-calculated cosine tables, and made use of the trigonometric relation:

cos(A)*cos(B) = [ cos(A+B) + cos(A-B) ]/2

All you need to to is scale the two large numbers to between 0 and 1, look up the scaled numbers in your cosine table to find the angles they represent, add and subtract the angles, find the cosines of the new angles in your table, and average them.

This speeds up multiplication by reducing a series of single-digit multiplications and additions to some table lookups, three addition/subtractions, and a division by two.  Before calculators, computation was much more of a scarce resource; anything that could speed it up was useful.  Before logarithms were invented, this could speed up your multiplication a lot.

For example, let's say we wanted to multiply 45678 x 23456.   Convert them to (0.45678 x 10^5) x (0.23456 x 10^5).  Combine the two multiplied 10^5 base 10 exponents into 10^10 and save it for later.  Looking up 0.45678 and 0.23456 in the cosine table says the angles representing them are 62.820 and 76.434 degrees, respectively.  Adding and subtracting these angles gives new angles of 139.254 and -13.614.  The cosines of these angles are -0.75761 and 0.97190.  Averaging them gives 0.10715.  Multiply this by the base 10 exponent of 10^10, and you have 1071400000. 

By my calculator, 45678 x 23456 = 1071423168, which to the correct number of significant figures, is 1071400000, the same answer from the cosine method.  Pretty neat, for the days before calculators.

Burton MacKenZie www.burtonmackenzie.com

[1] Greek "prosth-" or "adding"  and  "-aphaeresis" or "subtracting"

A Green Energy Business Model

I just recently heard about a new sustainable-power business model.  It would work best in areas where cost of electric power rates are in the medium to high range.  I hope it takes off. 

Customer: Warehouses, BigBox businesses, etc.  It could be any operation that consumes power (for lighting, machinery, etc) and has an open roof sprawl.

Technology: Solar panels on the roof collect cheap energy to reduce higher energy costs paid to local grid power company.  Solar power on the roof could supply approximately 20% of the electrical needs of the business.

Problem: Solar power installed on the roof would be expensive up front.  The power costs are only cheaper when amortized over decades.  It requires skilled maintenance, which is not the core competency of the business.

Solution: A third party business whose core competence is solar power purchases the equipment and maintains it.  The Big Box business signs a 20 year contract to buy all the solar power supplied from the roof by the 3rd party business.

Everybody's Happy: The Big Box business saves on energy costs.  The 3rd party power company makes money and employs people.  The citizens get cleaner power generation used in their environment.  Ubiquitous solar power will drive down cost, making it even more competitive.

I think this is the first time I've ever gotten excited about a business model.  Merry Christmas.

Burton MacKenZie www.burtonmackenzie.com

Net Neutrality: Internet ~ Electricity

My internet has been down for almost a week.  

It's been like turning off the electricity.  Sure, you can live without it, but it really makes a lot of things easier.

How long is it before our functional society is so deeply embedded into a bidirectional commodity data service (i.e. the internet) that it will collapse without it?  At one technological company where I worked, during an internet outage the apathetic network guy told the president he'd "get around to fixing it later".  The president pretty much flipped out on him, as lack of internet brought the productivity of the engineering design team to zero.  The network guy, of all people, did not understand this.

If a commodity data service can be comparable to commodity energy (e.g. electricity, natural gas, etc), why would you pay more for the end use of the service?  You do not pay a different base rate for electricity if you're using it for your stove instead of your laptop.  That would be crazy.  Similarly, you should not be charged differently depending on if your data is used for email, surfing, or downloading mp3s.  Data should be charged at a rate of amount of data actually transferred (i.e. how much of the commodity did you use), with other commercial factors like average bandwidth, latency, etc.

If the power company wanted to charge us different rates depending on the nature of what you were using it for, there would be revolt in the streets!  That is an artificial business model based solely upon squeezing more money out of the consumer, not based on costs of supplying the commodity.

That is what Net Neutrality is about.  Data is a commodity service, like electricity.  By allowing coporations to charge you based on the intended use of the data rather than proportional to the actual costs of supplying the service, they are imposing an artificial business model, for the good of them, not you.  Regardless of what their propaganda tells you, it is only about more money for the corporations supplying the data service.  There should be revolt in the streets if this ever happens.  Do not let it.  Data must be neutral!

Control the nature of the data service, and control the blossoming massively parallel interconnected society.  I do not believe corporations should have a grip on our balls.  Do you?

Burton MacKenZie www.burtonmackenzie.com