Engineer & Entertain

Ideas I grapple with

Archive for the category “math”

Spiraling into control

I am building a table for my Handibot as one of my biggest challenges thus far is registering the machine in the same spot when I move it to change bits. The table will have dog holes bored every 3 inches. The Handibot will have a fence and work pieces will be clamped against the fence via offset clamps.

The screw does not go through the center of the circle. It is offset so the circle rotates like a cam.

Rather than cut circles and mount them offset, I decide to make cams based on the Fibonacci spiral. I am not sure if there is any advantage or disadvantage to this. My reasoning was that a spiral would have a smoother transition in clamping. We’ll see how this works in practice.

I laid out a rectangle and spiral based on the Fibonacci numbers with a pair of dividers. Alternatives to using dividers/compass include:

  1. Multiplication – the 3″x3″ square is effectively the number 8 square in the picture below. To make the 5 square, multiply 3 by 5/8ths. The 5 square is 1.875″ x 1.875″. To make the 3 square multiply 3″ by 3/8ths and you will find the 3 square is 1.125″ x 1.125″
  2. Print a spiral. Adhere the paper to your work piece with spray adhesive and cut it out.

I laid out a 3″ x 3″ square on a scrap of 3/4″ thick plywood. I set my dividers to divide the square into 8ths.

The 3″x3″ square can effectively be thought of as the 8 square

From the outer corner of the 3″ x 3″ square I paced out five spaces. This is the corner for my 5 square.

Five little steps and I made the 5 square

I repeated this process to create the 3 square, 2 square, and 1 squares.

I used a compass with a pencil to trace the arc in each square. The compass legs are spaced the length of a side of a square.

Then I cut out a small rectangle from the 8 square. This provides a handle for me to grip when tightening the clamp.

Cut with a handsaw.

Next I used my fret saw to cut the spiral. I used a sander to sand down to the line.

Golden

I drilled a 3/4″ diameter hole in the spiral for a dowel. The dowel is 1.5″ long and glued in place.

Looks like a duck

The clamp does hold pieces against a fence, but I am unsure if it will be strong enough while using a CNC router. I’ll report back with those results eventually.

Today’s Google doodle is Turing heads

Today’s Google Doodle is in celebration of Alan Turing, a mathematician and computer scientist whose discoveries are still useful today.  He would have been 100 years old today had he not killed himself with a cyanide laced apple.  Turing made countless contributions to math, computer science, artificial intelligence, and aided in chemistry & biology research.

During WWII, Turing played in a park and showed he was da bombe by solving some German ciphers.  Aside: You can make your own Enigma machine out of paper here.  His Turing machine is a hypothetical device able to follow a procedure i.e. an algorithm. The Turing test gives us a basis to evaluate artificial intelligence.  He also studied plants’ predisposition to arranging their leaves in a Fibonacci sequence.  Despite solving ciphers, testing human-computer interaction, and papers on morphogenesis, Turing’s greatest contribution is solving the halting problem if only because it gave us this comic.

Turing’s work and life are utterly fascinating.  Who knows what other contributions he would have made had he lived to 100.  If you are wondering how to use today’s Google doodle see the video below.

Take A Number

An iPhone developer anonymously recorded iPhone users’ numerical passcodes.  Using statistical analysis i.e. mode and average, he created a list of the ten most common PIN.

Because I am concerned about unintentionally selecting a number or password with meaning I use a random number generator to create a PIN.  I still have a bias; however, because I dismiss numbers like 7777 or 1234 even thought they arise randomly.  The most common numbers are listed below.  Every phone, GPS, and garage keypad I come across will experience me entering these numbers.

  • 1234
  • 0000
  • 2580
  • 1111
  • 5555
  • 5683
  • 0852
  • 2222
  • 1212
  • 1998

Also of note, 1990 – 2000 were in the top 50 passwords and 1980 – 1989 were in the top 100.

Lightning Calculation Calendar

The gentleman who created the 2010 nomogram calendar has created a calendar for 2011 which will add in your ability to do lightning calculations.  If you have ever wanted to convince someone that you are a robot demonstrating your ability to perform rapid mathematical operations will certainly aid in that endeavor.

Nom Nom Nomogram

A variety of older technologies fascinate me, but I am particularly fond of slide rules.  I believe my fascination follows this course of reasoning: Suppose Y2K did shut down all of our computers and computer automated processes, with a slide rule we wouldn’t be sent back to the Iron Age, but rather the 1940s or 1950s.  If you ignore the second World War and fear of communism – those times did not seem so bad.  Among other things, slide rules were most helpful in landing on the Moon and returning a crippled space craft safely.  Quite simply, I am impressed by the wide range of what a slide rule can do.

I have recently become intrigued with another analog computation device called the nomogram.  A slide rule is a general calculator, while the nomogram is a graph which provides a quick answer from a complex calculation.

In nursing & medicine we use nomograms frequently for tracking and calculating weight & height & BMI, medicine administration, and more.  I was not aware of their varied calculations until I came across this 2010 calendar and a website, Dead Reckoning, featured on Make.  The blog is very fascinating and I can only hope there is a 2011 calendar featuring elegant-but-lost math.  I’m tired of looking at my pin-up girls calendar.  That’s not true.

The Prisoner of Benda Therom

Any time a sci-fi show runs into a problem there is usually some manipulation of the known world which solves the problem.  Typically, the solution is a completely wrong extrapolation of something we learned in high school science with jargon flying left and right to misdirect and make it sound creditable (I’m looking at you, Eureka).  When this occurs geeks and nerds across the Internet get on forums and discuss the unrealistic minutiae while being content with the Moon actually being blown off its orbit instead of being blown apart.

I digress, a writer of Futurama decided to make a mathematical proof of the plot point so as to not remove its viewers from a suspension of belief.  You can read more about it here.  You can also get a clear view and explanation of the therom here.

*Jamie & Paul, should I have said therom when I meant proof or proof when I meant derivative just imagine it was a typo.

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