Mathematics :

Shop Math

Every craft has its own mathematical needs and it is important for the worker to be able to handle those relatively simple needs. Often this includes a working knowledge of basic geometry and trigonometry. Solving for volumes is needed to determine material needs in almost every field and basic solutions to triangles are needed to plan and build almost any structure.

No matter how well we did at these subjects in school most of us cannot remember all those formulas. This is what reference books are for. One of the best for shop math is the book that I repeatedly recommend to all metalworkers and that is MACHINERY'S HANDBOOK by Industrial Press.

The math section of this long published reference has changed very little over the years. The things I use the most are the solutions to right triangles and volumetric equations. It also includes beam stress and deflection formulas, centroids or centers of gravity and other standard engineering subjects.

In the 1980's I wrote a mass volume calculator program for the PC called Mass2. It was written to ease the repetitive task of calculating weights of machine components. Much of its core mathematics is from MACHINERY'S HANDBOOK. The on-line version is called Mass3j and is linked below.

Mass3j calculates the volumes of cylinders, rectangulars, right triangular prisms and polygonal prisms. Weight in various materials is displayed. It also has links to our other math articles and conversion functions below.

Laying out ovals by the string and pin method. Formulas and calculator.

Laying out cones to make chimney caps, funnels etc.

Punching Force
Chart and calculator with punching force in tons for round hole or blanking.

Riveting Force
Force necessary to hot rivet steel rivets in a press.

Common English metric conversions. Inches to Millimeters, Feet to Meters, Fahrenheit to Celsius, Pounds to Kilograms, PSI to k/Pa (Pounds per square inch to kilo Pascals), lb/cuin to g/cm3 (density), HP to KW, Square Feet to Square Meters, Cubic Inches to cm3, cubic feet ro cubic meters.

Layout and Squares
iForge demo on using framing squares, truing your square and magic triangles, stock gages.

Bells, Gongs and Triangles
Math and Design of Simple Vibrators, The 12th Root of Two, Binary series.

English Hundredweight Conversion
Calculates to and from the English Hundredweight system. Includes rules.

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Numeric Series and Mathematical Thought: Twelfths have a problem in a simple fractional series but there are more complex series. The simplest and most thought friendly is also accurately produceable with simple tools (divider or compass and straight edge) is 1/2, 1/4, 1/8, 1/16. . . The ancient Greeks had rules for mathematics that said that every true solution must be able to be defined with those simple tools. This created the problem of "squaring the circle" (defining PI) using a compass and straight edge in a finite number of steps. There has never been a solution to this problem even though it has been worked on for over 2000 years (The History of PI by Petr Beckman).

In twelfths the 1/4 = 3/12 presents the problem of division by thirds. Using simple tools this means iterating (trial and error) which can come close but is never perfect. Tenths is the same requiring division by fifths immediately after halves (5/10). Division by 5ths is even more difficult than thirds. The metric system using tens is a good numeric system for monetary purposes and counting but is lousy for measuring systems. This is the fallacy of the metric system of tens being better.

The ancient Summerians, the folks with the clay tablets and cuneiform writing, had one of the earliest known mathematical systems. It used base 60. This was symbolic of their year which had 360 days (whence comes our degrees in a circle). They believed that the world must be mathematically perfect and thus 60 and 360 were very important numbers to them. Of course their calender quickly failed so they tossed in a few holy days that didn't count and kept the system going for a little longer. . . Religion met facts early and failed the test.

Later base 10 (which we use today) was universally adopted for those of us that have to count on our fingers and toes. However it did not become widely used until the invention of zero, again in the Middle East.

However ever since the time of the Summerians we still use base 60 in everyday life (besides measuring angles and therefore navigation). What else do we use it for? Telling time.

Here is where mathematical bases and twelfths come into play. Sixty is a wonderful number and many bull gears in lathes have 60 teeth so that it can be used for dividing parts. More even and odd fractional numbers go into sixty than any other number.

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 30, 60

The fact that the first six is a continuous series had to have intrigued the ancient Summerians. Six also gives you a series with 6, 12, 24, 32, 30, 36, 42 . . . all numbers that can be created by the division of a circle with a compass starting with a hex which also creates equal lateral triangles. And this brings us back to twelfths because we can evenly divide a circle into 3, 6, 12. . with a compass. Cut and straighten the edge and you have those odd numbers created with even divisions. . .

The metric system ignores all this wonderful mathematical history and has not been fully adopted in ANY place (country, business, school). Time is still measured using the ancient Summerian system (60 min, 12 hours, 1/4's and 1/2's) and angles measured using their 360 degrees divided into common fractions of 180, 90, 45.

Officially the metric system uses radians where the circle equals 2π (2pi). Computers use this system for some very complex reasons as well as meeting the metric standard. However, this results in every computer program dealing with angles and trigonometry needing to convert degrees to radians using the conversion factors 180/PI and PI/180. The system also leaves us back in the world of fractions where we would use π/2, π/4, π/8 (180°, 90°, 45°). Then there is π/3 = 60°, π/6 = 30°, π/12 = 15° and we are back to the ancient Summerian system. . . AND FRACTIONS.

In the early days of the metric system various divisions of the circle were tried including the grade of 400 units so 100 grade = 90°. The problem was that the circle cannot simply be divided into 10 (unless you use 40 grade) and the system fails mathematically because no even tens division of a circle makes a right angle or a straight line. . . The grade still lurks around and is the reason we stopped using centigrade for measuring temperature (1/100th grade = 1 centigrade). Yeah, the Summerians win again.

Mathematical literacy (Numeracy) including fractions is easy to teach but our schools have always waited too long to introduce fractions (a three year old understands HALF a candy bar). They also do not teach the history of math so almost nobody knows why there are 12 hours on the clock or 360 degrees in the circle (including most teachers). . . much less WHY the conversion between Fahrenheit and Celsius (centigrade) is -32 * 5/9.

Personally knowing the REASONS things are makes it easier for me. Take the Fahrenheit and Celsius problem above. The difference between the ice point (32°F) and boiling point (212°F)in Fahrenheit is 180° (like half a circle). Fahrenheit set it up that way. In the metric Celsius scale it is 100°. The ratio of 100/180 reduces to 5/9. Since Celsius starts with the ice point at 0° and the Fahrenheit at 32° you have to add or subtract 32 to start.

IF these simple things had been taught in school it would have been much easier to remember F -32 * 5/9 = C.

Every mathematical system you can imagine has been tried and each has some advantage or disadvantage. The computer geeks say that if we had been born with 4 or 8 fingers it would have made the computer revolution much easier OR much earlier. Base 8 (octal) and 16 (hexagesimal) convert directly to binary (on/off high/low) and are what all computer math operate on at the chip and programming level.

Twelth Root of two  Binary series.  Graphic Copyright (c) 2001 Jock Dempsey
We also apply binary number series to music as well as 12 notes in an octave (8 full and 4 half). This results in the curious formula for Western scales and fret spacing of the reciprocal of the twelfth root of two to the power of n. . .

More 12ths, binary series which are reciprocals of 1/2, 1/4, 1/8 . . . around and around the circle we go.

Numeracy dictates that we understand orders of magnitude and that even using the smallest common measurement units we can define the entire universe in a range 3 significant figures and two digits of magnitude (in almost any mathematical system). That and a little history go a long way. .

   - guru - Saturday, 10/01/05 11:17:20 EDT

References and Links

Copyright © 2004 - 2006 Jock Dempsey,

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