Sonar

How the Navy Hears Subs 3,000 Miles Away. (It's Genius)
Second Order Science

Previous related posts:

• SOSUS - my run-in with SOSUS in college
• Star Trek Science - submarines so quiet they can't hear each other

Wikipedia:

• SOSUS
• SOFAR Channel

Vampire Numbers

Vampire Numbers - Numberphile
Numberphile

Dennis sent me this video, the idea being that searching for Vampire Numbers might be a good use for the factoring scheme I'm working on.

Velocity & Distance

Demos Graph

I'm trying to work out some elementary math problems, but for some unknown reason I haven't been able to actually work on them, so I thought I would just do a simple math problem, just to get moving.

This problem is: How does velocity change with distance?

Start with old distance from time formula.

Set acceleration to 5 so we get a nice curve.

Old velocity from time formula

Solve the first formula for time

Substitute our new value for time into our velocity formula

Assign distance to x and that gives us the graph we have above.

If we view this graph as a car accelerating at a rate of a low 1/6 G, then by the time it has gone 80 feet (or six car lengths) it will be going pert near 30 MPH.

RIP Chuck Norris

Chuck Norris Does Math

Via MOTUS A. D.

Late to Pi Day

Computing Pi For Pi Day!
Dynamically Typed

YouTube blurb:

Using my 92 year old Walther RKZ mechanical calculator to calculate Pi. 355/113 is a very good approximation for the famous constant. It's a common example for mechanical calculators but it is still fun to see.

Elastic Knots

Elastic knots are really mind bending
Steve Mould

Dennis brought 3 or 4 similar puzzles to lunch the other day. I can deal with bandsaw blades, but I was afraid to tackle anything more complicated because I might screw up and put in a kink it. Figured for sure I could look online when I got home and find a bunch of these things, but no such luck.

Einstein Tile

Einstein Tile

Discovery of Elusive ‘Einstein’ Tile Raises More Questions Than It Answers

David Smith, a math hobbyist in Yorkshire, England, has discovered a 13-sided shape that eluded mathematicians for decades. The craggy, hatlike shape is called an “einstein,” based on the German for “one stone.” If you used einstein-shaped tiles to cover your bathroom floor—or any flat surface, even if infinitely large—they would fit together perfectly but never form a repeating pattern. For decades mathematicians have been hunting for tile shapes like these that can form only nonrepeating arrangements, called aperiodic tilings. They started with sets of many different tiles: the first set, discovered in 1964, required 20,426 distinct tiles, which was later simplified to 103. By 1974 mathematician Roger Penrose had found two tile shapes that, when combined in a mosaic, never formed a repeating pattern.

Someone ought to make some jigsaw puzzles using this pattern. Shoot, someone ought to make Einstein floor tiles.

Break Einstein into two words (Ein stein) and Google says it is 'a stone'.

Via Dennis

Fun with Algebra

Harvard University Admission Entrance Tricks | Radical Algebra Aptitude Test
Learncommunolizer

There are several YouTube channels with people explaining how to solve various math problems. I enjoy working simple math problems, it's good exercise for my brain. Sometimes, if the problem looks like something I can solve, I will attempt it. Some look like they are way above my pay grade, I ignore those. This one looked like I ought to be able to handle it. My first instinct was to expand the expression and hope that a solution would magically appear. They do, sometimes. But the power of ten meant that the expansion was going to be huge, and I don't want to do that, it would take an awful lot of scribbling. Maybe this guy has a trick up his sleeve, so I watched it. I have got to say his method is pretty clever.

Fun with Numbers

A couple of word problems told at Longbottom Coffeehouse today. The videos are not directly related to the text. 

Medicine

Skills Video: Programming an IV Pump
Student Nursing Organization at UNH

A nurse needs to administer 115 milligrams of a medicine intravenously over a period of 75 minutes. The medicine will be administered through a hypodermic needle via an IV line using a calibrated pump. The pump wants the rate in milligrams per hour. What do you set the dial to?

The problem can be taken as a proportion: X milligrams over 60 minutes is proportional (equal) to 115 mg over 75 minutes:

Multiply both sides by 60. 60 over 60 on the left side cancels out leaving X by itself. Notice that 60 and 75 have a common factor of 15, reduce that and you are left with 4 over 5, which is the same as .8. Multiply that by 115 and you get 88 plus 4.0 which comes out to 92. Closest setting on the pump is 88, so that's where you set it.

Driving

Acura TL 60-0 mph Braking Test
Born to Drive

You are driving down the freeway at 60 miles per hour following the car ahead of you at a distance of two seconds. (That is, you notice something by the side of the road that you can use as a marker. When the car ahead of you passes the mark, you start counting, one-thousand one, one-thousand two, until you reach that same marker.)

Trailing two seconds behind means you are 176 feet away. (60 MPH is equivalent to 88 feet per second, 88 times two is 176.) The question is - if the car you are following were to suddenly turn into a brick wall, would you be able to stop before you hit it?

For this I am going to pull out two of my favorite formulas:

• V = AT (Velocity equals Acceleration times Time)
• D = (1/2)AT^2 (Distance equals one-half times Acceleration times Time squared)

We know the Velocity (88 FPS) and the Distance (176 Feet). Will this be enough to solve this equation? 

Substituting Velocity for AT in the second equation will give us	D = (1/2)VT
Substituting what we know for Velocity and Distance we get	176 = (1/2)88T
Multiply	176 = 44T
Divide both sides by 44:	4 = T

So if we can brake to a stop within four seconds, we will not hit the wall, the brand new wall that just suddenly appeared in the middle of the freeway.

Okay, how much strain is that going to cause? And will our car be able to do that?

Go back to our old friend V=AT and plug in our Time and Velocity and solve for Acceleration, we find that we need to slow down at a rate of 22 feet per second per second, which is like two-thirds of one gravity, so it might be possible. A race car would certainly be able to do, but how about my Korean sedan? Google says:

When it comes to the 2008 Hyundai Sonata's braking distance, specifically from 60 mph, Edmunds review indicates it is quite good for its class, coming in at less than 130 feet.

So we should be good, assuming we are paying attention, our reactions are on a hair trigger, the road surface is free of any kind of friction reducing substances, like ice, oil, dust or debris. So, in the real world, if you wake up and get your brake applied in time, you will probably hit the wall, but you probably wouldn't die in this completely fictional situation.

Probability or The Story of Google

What are Markov chains? And why are they so useful?
Veritasium

Does a good job of explaining how Google works, which is remarkably simple, all it takes is a bigger computer. If you are just looking at letters, you just need a simple array with room for 26 entries. If you are looking at words, you need an array with a zillion entries, because not only do you are you using all the English words, you're also using all words from all the other languages and all the names and acronyms. And numbers, and punctuation, and symbols. And then, after you have counted all the words and populated your data base, you need to be able to answer a zillion queries simultaneously, which I suspect means you need to make multiple copies of these data bases. No wonder they're building all these data centers in my neighborhood.

And do not forget that all of this is being paid for by advertising. A tenth of a cent times a billion clicks is still $100K, and that probably happens, what? Hourly?

A couple of people mentioned:

Masayoshi Son (1957 - )
Japanese Businessman

Andrey Andreyevich Markov (1856 – 1922)
Russian Mathematician

Pavel Alekseevich Nekrasov (1853–1924)
Russian mathematician

On the downside, he drops in a couple of lines about global warming. I don't know if global warming is a real problem, but I do know that all of the measures western environmentalists have proposed are worse than useless.

It ends with an ad for Brilliant at the 31 minute mark.

No, you can't get there from here

This mechanism shrinks when pulled
Veritasium

Curious stuff. Wikipedia has a page about Braess's Paradox. 

The video has an ad that runs from 12:20 to 14:15.

Via California Bob.

Hillbilly Elegy

In that same bookshop where I read that bit about baseball errors, there was also a box of free books for 'burning or your rage room'. Well, I gotta see what we have here, and what do I find? Several copies of J. D. Vance's Hillbilly Elegy along with several Harry Potter books. Remember this is northwest Portland, a very deep blue corner in a very blue state. I picked up Hillbilly Elegy and took it with me. We already have all the Harry Potter books.

I am about half way through, and so far it's been his life growing up amongst the hill folk, and a rough bunch they are. It's kind of amazing that he amounted to anything, much less Vice President of the United States.

Anyway, on page 58 I came across this little bit that I thought was entertaining. I mean, I thought I was a math whiz:

Alongside these conflicting norms about the value of blue-
collar work existed a massive ignorance about how to achieve
white-collar work. We didn't know that all across the country -
and even in our hometown - other kids had already started a
competition to get ahead in life. During first grade, we played a
game every morning: The teacher would announce the number of
the day, and we'd go person by person and announce a math equa-
tion that produced the number. So if the number of the day was
four, you could announce "two plus two" and claim a prize, usu-
ally a small piece of candy. One day the number was thirty. The
students in in front of me went through the easy answers - "twenty-
nine plus one," "twenty-eight plus two," "fifteen plus fifteen." I
was better than that. I was going to blow the teacher away.

When my turn came. I proudly announced. "Fifty minus
twenty" The teacher gushed, and I receive: two pieces of candy
for my foray into subtraction, a skill we'd learned only days
before. A few moments later, while I beamed over my brilliance,
another student announced, "Ten times three." I had no idea what
that even meant. Times? Who was this guy?

93

How Europeans say 93
Roya

Denmark's explanation is clear as mud, so I tried to figure just what they saying and I think I have it:

Three plus (five minus a half (which is four point five) and times that by twenty)

3+((5-0.5)*20) = 93

Count Like An Egyptian

The Bizarre Fractions of Ancient Egypt
Wrath of Math From Wikipedia:

The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057, pBM 10058, and Brooklyn Museum 37.1784Ea-b) is one of the best known examples of ancient Egyptian mathematics.

It is one of two well-known mathematical papyri, along with the Moscow Mathematical Papyrus. The Rhind Papyrus is the larger, but younger, of the two.

In the papyrus' opening paragraphs Ahmes presents the papyrus as giving "Accurate reckoning for inquiring into things, and the knowledge of all things, mysteries ... all secrets". He continues:

This book was copied in regnal year 33, month 4 of Akhet, under the majesty of the King of Upper and Lower Egypt, Awserre, given life, from an ancient copy made in the time of the King of Upper and Lower Egypt Nimaatre. The scribe Ahmose writes this copy.

Several books and articles about the Rhind Mathematical Papyrus have been published, and a handful of these stand out. The Rhind Papyrus was published in 1923 by the English Egyptologist T. Eric Peet and contains a discussion of the text that followed Francis Llewellyn Griffith's Book I, II and III outline. Chace published a compendium in 1927–29 which included photographs of the text. A more recent overview of the Rhind Papyrus was published in 1987 by Robins and Shute.

Farther down the Wikipedia page is this image:

Ancient Egyption Units of Measurement in the Rhind Mathematical Papyrus

The image on Wikipedia contains some text, but the lines are very long which makes it difficult to read. Printing the image large enough to make the text legible requires two sheets of paper. I used Windows Paint. I had completely forgotten about this program, but it is around, at least on my Windows computer.

Anyway, I thought the text in this image ought to be converted to text, so I used OCR Online to extract the text and then edited it to correct the errors. The only words OCR had trouble with were the  highlighted words in color. Other than that it was perfect. Then I went ahead and chopped it up to make it more readable, at least in my mind. Here it is:

The present table is a concordance of ancient Egyptian units of measurement which are used throughout the Rhind Mathematical Papyrus, an ancient Egyptian document which is a record of elementary mathematics. The papyrus consists of four sections:

1. a title page with historical information,
2. a table of fractional calculations for 2/3 - 2/101, or the "2/n table" (where n is always odd),
3. a much smaller table of fractional calculations for the nine fractions 1/10 - 9/10, or the "1-9/10" table, and finally
4. a series of 91 "problems" or numbers, which are numbered from 1-87 (a modern convention imposed on the document to differentiate the problems) and include four additional items designated by moderns as 7B, 59B, 61B, and 82B.

Many of the latter problems make use of certain ancient Egyptian units of measurement, whether of length, volume, time, or otherwise, and this is what the present table summarizes. It happens that none of the 2/n table, the 1-9/10 table, or even most of the first several problems 7, 7B, 8-34 make any mention of units, and they are therefore not included in this table.
Almost all usage of units of measure is confined to the later problems in the papyrus (with some early exceptions, also listed), being the title page, problems 1-6, and problems 35-87, which include three additional items designated as 59B, 61B, and 82B.
For the title page and numbers 86-87, the context of the usage of units is not mathematical, but historical.

The units are grouped by unit type, and color-coded. From top to bottom, the unit types are:

• length,
• area,
• volume (deben),
• monetary (sha'ty),
• trigonometric (seked),
• food/manufacturing (pefsu), and finally
• "foodstuff" (loaves, des-measure).

The seked is not strictly speaking a trigonometric item, but its context is so close to our modern understanding of trigonometry that we identify it with that word.
The pefsu is also a kind of derived unit of measure relating to food preparation and manufacturing.
Finally, the "loaf' is not really a standard unit of measure within the papyrus as such, but it is mentioned so frequently in related contexts that it merits its own entry. In any one problem, it can be interpreted as a measure of solid food.
Likewise, the "des-measure", mentioned in only a few problems, can be interpreted as a volume unit of liquid measure (especially in the context of food and drink) which is not immediately related to the other volume units.

Entries in black indicate that a given unit of measure is explicitly named or entailed in the problem, in the original document.
Entries in gray indicate that although the unit's word is not expressly stated in that problem, the context of the document makes clear that the unit is implicitly being used in the course of the problem's calculation, statement, etc.
Note that neighboring groups of problems often tend to entail similar units, although in some cases the units are merely strongly implied by context, as opposed to being explicitly stated in the original document.

In the Rhind Papyrus, units of a given type have precise, exact, simple relationships to one another, which conversion factors we can and therefore do easily express in modern terms, using only integers.
Although there is historical evidence that certain units of measure can be "concretely" compared with modern units, these exact specifications are not essential to reading the Rhind Papyrus as a mathematical document.
Indeed, one can read the papyrus has having an internally consistent system of units of measure, which is all that is presented in this table and explanation.
The reader is cautioned that ancient Egypt had a very long history, and that the present information about units pertains directly to the Rhind Papyrus only.
At other periods of ancient Egyptian history, certain units were taken to have different conversion factors relative to other units.

Among units of length, 1 khet = 100 cubits = 700 palms = 2800 fingers.
Among units of area, one square khet is called one setat, and 1 setat = 100 cubit strips, where one cubit strip is a rectangular strip of area being 1 cubit by 100 cubits (or any other sector of equal area).

Among units of volume, the following holds:

2 cubic cubits = 3 khar = 60 heqats = 600 hinu = 19200 ro.

Furthermore as one might expect, the somewhat redundant "hundreds/multiples" versions of the heqat and the ro can be equated with those same "base units" in the following wise:

1 hundred quadruple heqat = 2 hundred double heqats = 4 hundred heqats = 100 quadruple heqats = 200 double heqats = 400 heqats = 128000 ro = 64000 double ro = 32000 quadruple ro.

Combining these two chains of equalities, rearranging their terms, and expressing all of the units' conversion factors in the simplest terms of integers yields the following comparison among all of the standardized units of volume measurement in the Rhind Papyrus:

3 hundred quadruple heqats = 6 hundred double heqats = 12 hundred heqats = 300 quadruple heqats = 600 double heqats = 1200 heqats = 40 cubic cubits = 60 khar = 12000 hinu = 384000 ro = 192000 double ro = 96000 quadruple ro.

Among time units, since the Egyptian month is always exactly 30 days, then 6 years = 73 months = 2190 days. The remaining units, being of different types and not having simple or relevant conversion factors into other units of like type (at least as far as the papyrus itself is concerned) are therefore not expanded upon here.

More posts about fractions.

Another post about ancient mathematics: the Babylonian Plimpton 322 clay tablet. It dates from 1800 BC, so roughly the same era the Rhind Papyrus.

Pie Are Square

Get Fuzzy

More from today's funny papers. The business of Lowe Tech awarding an honorary degree is a distorted version of reality. There is a guy who was able to quote the digits of Pi for several hours:


The man with the fantastic brain - a Savant
011BEK

Wikipedia has a page about this guy - Daniel Tammet.

The Sunday Comics page in The Oregonian changed about a month ago. A month? Maybe longer. Anyway, they are using different paper for the comics than the regular news. The paper is thicker and narrower. Disconcerting the first time you pick it up. They've also changed the selection of comics and the layout. That rabid left-wing comic Doonesbury used to be on the front page, it has been moved to the interior. Stephen Pastis's Pearls Before Swine has disappeared. The ancient, stupid, comic, B. C. is now at the top of the front page.

Fun with Numbers

Math puzzle – Can YOU find the area?
Math Queen

In my cave, this qualifies as fun. Looking over the videos on display on YouTube, I noticed this one. It looks like a fun little problem, and it was. When I am tired, usually from not getting enough sleep, I'm looking for some mindless entertainment. One of the games listed in the sidebar are my usual choice, but simple geometric / algebra problems can also amuse me for a few minutes. This one didn't take long, but, as usual, I can't tell you how long because once I started working on it time stopped. Below is a shot of my paper with my work. The answer to the problem (why would you care?) is 2.5 pi.

Permutations

Factoring and Permutations Spreadsheet

Talking at lunch Tuesday*, Dennis mentioned he was playing with numbers, specifically looking for interesting patterns in permutations of a four digit number. Why? Because it's there, that's why.

I got to wondering if you could factor a number using a spreadsheet. Since we are limiting ourselves to four digit numbers, there will be no factors greater than 100. There are only 25 prime numbers under 100, so it shouldn't be too bad.

The spreadsheet is full of formulas, but I only had to figure out a couple of them and then copy and paste them to make the whole sheet.

The spreadsheet is not very smart. The largest number of factors a 4 digit number can have is 13. Two to the 13th power is 8192. Add a 14th factor and the product will have 5 digits. So 13 is the largest number of times we will have to divide.

We simply divide our starting number by our chosen factor and then count the number of times the division came out even. Our quotient becomes the starting value for the next line and the next factor.

Next I tried my hand at generating permutations. This one didn't work out quite so well. I ended up generating the permutations by hand (there are 24 permutations of 4 character strings) and then using those values to extract digits from the original 4-digit number.

Today I worked on a way to generate permutations automatically. It works, but criminently, it took a bunch of equations. 

I used string functions to cut up the number which means it should work for any four characters. It could probably be done using mathematical operations. I don't know if that would be any simpler.

*Tuesday three months ago.

Money for nuthin and your chicks for free

Big Couch

If you have ever tried to move a couch around a corner in a hallway, you know what a pain it can be, especially if it's heavy. I've done it a couple of times when my kids moved into apartments. We had to stand the couch on end to get it in. I remember a place in Eugene where the entrance door opened onto a hallway that ran sideways, so three feet inside the door was a wall. Turning the couch on end, it was too long to fit through the door, so we had to slide one end through the doorway and then tilt it up while sliding the lower end through the doorway. Now it is standing up on end inside the apartment. Then we turned it 90 degrees in place and reverse the process and slide the lower end through the doorway from the hallway into the living room. Giant pain.

Anyway, once upon a time, some dude wondered what the biggest shape that could be moved around a corner in a hallway. 30 years later a student answered the question with a 120 page paper. It's an entertaining story.

Money For Nothing
Dire Straits

P. S. The post title comes from this song which includes these lines:

We got to move these refrigerators
We got to move these colour TVs

It comes to mind every time I have to move something big.

Geometry Refresher

Brilliant Geometry Puzzle
Andy Math

I am a sucker for math problems. I like the ones that are easy for me to solve, I don't much care for the really esoteric stuff, i.e. problems I can't easily solve. I started watching this video and I'm thinking I should be able to solve this. I attack with the Pythagorean Formula (which is basically the only tool in my geometry toolbox) but I get nowhere. So I resume watching to see how he is going to solve it and he immediately brings up the Intersecting Chords Theorem. The what? Never heard of it. What the heck is this thing? So, more YouTube:

Intersecting Chords Theorem! (explanation and examples)
You Can Learn Math with Alyssa

I like Alyssa, she moves a little too slowly for my taste, but she is crystal clear and she has a pleasant voice. But then she comes to Central Angles and Inscribed Angles and my brain objects, so more YouTube:

Central Angles and Inscribed Angles! (theorems AND examples)
You Can Learn Math with Alyssa

Alyssa explains what the theorem is and shows how to apply it, but she doesn't explain how the theorem is derived, what some people call 'a proof', so now we need another explanation.

Inscribed angle theorem proof | High School Geometry | High School Math | Khan Academy
Khan Academy

Now that I think about it, I probably learned about all this back in high school, but I think that was it for geometry. After that is was calculus, trig and matrices, and geometry kind of got left in the dust.

Fun with Math

United States of Voronoi

California Bob reports:

I just learned about this. This is one of those things that's easy to see but difficult to describe.

Voronoi Cells

A Voronoi diagram partitions a plane around points or "seeds" within it.  It THINK it divides the area so that the boundaries are always halfway between one seed and the next closest seed.

The official description is, boundaries are drawn so that all points within that boundary, are closer to its seed, than to any other seed.

If each seed was a repelling magnet, would this describe the same boundaries?

Here [image at top] it's used to draw state boundaries around state capitals.