An Error in Reference Angle Post

I've found a problem with the procedure for finding reference angles.  According to the process, when you get a value between but not including 90° and 100°, you repeat the loop.  But, the loop doesn't handle values between 90° and 100°.  So, I have to change the step at the end that says:

if measure is greater than 90°, perform procedure again using new value of measure

into the steps that say:

if measure is greater than or equal to 100°, perform loop again using new value of measure

and then:

if measure is greater than 90°, subtract measure from 180°

The last step stays the same:

ref is measure

Errors Found in Previous Post

I have found a couple of errors in my first post on 20 April 2010 that reads, in part,

To define a point on a plane, you specify two values on each of two axes out of a possible three.  This contrasts with the Cartesian coordinate system, where you would pick two axes out of a possible of four axes....

It should say something more like,

To define a point on a plane with my coordinate system, you specify one value on each of two axes out of a possible three.  This is different from the Cartesian coordinate system, where you would pick one axis out of a possible four, and then another out of a possible of two....

I think I prefer the other method of locating points that is used further in the post.  It says that you can pick regions, or quadrants in the case of the two dimensional Cartesian coordinate system, and then specify them with appropriately sized bits of information.

The Strengthative Operator

a + b = c

a · b = c

where c is equal to a added to itself b number of times, like this

a + a + a ... = c

a ^ b = c

where c is equal to a multiplied to itself b number of times, like this

a · a · a ... = c

a strength b = c

where c is equal to a raised to the power of itself b number of times, like this

a ^ (a ^ (a ...) ) = c

Unfortunately, the character set does not include the symbol that I made up for the strengthative operation.  Here is a drawing of it.

A Problem with 60 Degree Angle Trigonometry

I have rediscovered a problem with 60° angle trigonometry. The problem is that arccos(a) and arcsin(b) for this trigonometry are not functions. In the following drawing, A and C are points on a unit circle with center B. θ is the measure of the angle ABC. The measure of the angle CAB is 60°. The value returned by arccos(1) would validly be both zero and θ, as in the drawing. Similarly, the value returned by arcsin(1) would validly be both θ and 120°.

Quick Procedure for Calculating Reference Angles

Near the beginning of my 11th grade of high school, in Mr. Hopkins's Trigonometry/ Analytic Geometry class, I noticed that the reference angle of an angle measure like 200° is 20°, and it is the same as the reference angle of 2 million degrees and 200 million degrees and 2 trillion degrees, and so on.  I also noticed that it was similarly easy to find the reference angle of any degree angle measure that is a positive integer power of ten multiplied by 200°, 400°, 600°, or 800°.  Probably within a couple of weeks, I had developed a procedure (more of a programming solution than a mathematical one) for finding the reference angle of any degree angle measure with only addition, subtraction, loops, table-lookups and conditionals.  The simplest procedure with a calculator that I know is to divide the angle measure by 180°, then take the remainder and multiply it by 180°.  If it is greater than 90°, then subtract it from 180°.  Whether or not you have to perform that subtraction, the result is the reference angle of the angle measure.  Here is my procedure, which is far different.  I know, it doesn't look very polished.


digit      table(digit)
------     ------
 0          0
 1          -80
 2          20
 3          -60
 4          40
 5          -40
 6          60
 7          -20
 8          80
 9          -0


To find the reference angle measure in degrees (ref), start with the leftmost digit (digit) of the degree angle measure (measure).  While you are working with a digit in the hundreds place or greater, look up digit on the above table (table) and add the value to a temporary variable (temp) that starts with the value zero.  Continue adding to temp the remaining values corresponding to the digits in the hundreds place and greater.  When the tens place is reached, add the remaining amount of measure without looking up on table.  Set measure to the absolute value of temp, in degrees.  If measure is greater than or equal to 90°, repeat the procedure again.  If not, ref is measure.  The following is the procedure, but made more concise.


measure = given value in degrees
temp = 0
loop number of times equal to number of hundreds place or greater digits
  temp = temp + table(digit)
temp = temp + remaining digits
measure = absolute value of temp in degrees
if measure is greater than 90°, perform procedure again using new value of measure
ref is measure


Examples:
  ref(200°)
    measure = 200°
    temp = 0
    temp = temp + table(2) , which is 20
    temp = temp + (remaining digits , which is 00) , which is 20
    measure = abs(temp) in degrees , which is 20°
    is measure > 90° ? , which is no
    ref = measure , which is 20°


  ref(2200°)
    measure = 2200°
    temp = 0
    temp = temp + table(2) , which is 20
    temp = temp + table(2) , which is 40
    temp = temp + (remaining digits , which is 00) , which is 40
    measure = abs(temp) in degrees , which is 40°
    is measure > 90° ? , which is no
    ref = measure , which is 40°


  ref(3200°)
    measure = 3200°
    temp = 0
    temp = temp + table(3) , which is -60
    temp = temp + table(2) , which is -40
    temp = temp + (remaining digits , which is 00) , which is -40
    measure = abs(temp) in degrees , which is 40°
    is measure > 90° ? , which is no
    ref = measure , which is 40°


  ref(3090°)
    measure = 3090°
    temp = 0
    temp = temp + table(3) , which is -60
    temp = temp + table(0) , which is -60
    temp = temp + (remaining digits , which is 90) , which is 30
    measure = abs(temp) in degrees , which is 30°
    is measure > 90° ? , which is no
    ref = measure , which is 30°


  ref(3001°)
    measure = 3001°
    temp = 0
    temp = temp + table(3) , which is -60
    temp = temp + table(0) , which is -60
    temp = temp + (remaining digits, which is 01) , which is -59
    measure = abs(temp) in degrees , which is 59°
    is measure > 90° ? , which is no
    ref = measure , which is 59°


  ref(88888800°)
    measure = 88888800°
    temp = 0
    temp = temp + table(8) , which is 80
    temp = temp + table(8) , which is 160
    temp = temp + table(8) , which is 240
    temp = temp + table(8) , which is 320
    temp = temp + table(8) , which is 400
    temp = temp + table(8) , which is 480
    temp = temp + (remaining digits , which is 00) , which is 480
    measure = abs(temp) in degrees , which is 480°
    is measure > 90° ? , which is yes
    temp = 0
    temp = temp + table(4) , which is 40
    temp = temp + (remaining digits , which is 80) , which is 120
    measure = abs(temp) in degrees , which is 120°
    is measure > 90° ? , which is yes
    temp = 0
    temp = temp + table(1) , which is -80
    temp = temp + (remaining digits , which is 20) , which is -60
    measure = abs(temp) in degrees , which is 60°
    is measure > 90° ? , which is no
    ref = measure , which is 60°


If you notice any errors, please tell me of them.

First post about 60 Degree Angle Trigonometry

Decades ago, I tried to adapt 90° angle trigonometry to 60° angle trigonometry for use in the Minimal Axes Coordinate System, with some success.  However, I got rid of all of my progress several years ago.  I might work on it again sometime.  The idea was to go to the foundation of the math, including identities, and not just rewrite the tables.  Just as I write this, I am tempted to just rewrite the trigonometric function tables.

Three-dimensional Minimal Axes Coordinate System

A while back, I constructed a physical model of the 3-D variety of my coordinate system, because I was unable to imagine how the space was shaped.  I thought it might have been an icosahedron, a 20-sided figure.  Imagining the 2-D variety is easy.  Even if it is not, it could be drawn.  The shape is a hexagon.  Here is a drawing of the 2-D variety.

The shapes of the areas and spaces within the 2-D and 3-D varieties of the Cartesian coordinate systems are easy to imagine as well.  The 2-D one is a square and the 3-D one is a cube.  Imagining the space within the 3-D variety of my coordinate system is not as easy.  Here are three photographs of the 3-D variety.  The four bundles of like light-colored fuzzy sticks (white, pink (which looks white), yellow and orange) can be thought of as representing the four axes of the coordinate system.  If I call the four light-colored axes-vectors a, b, c, and d, then the vectors a+b, a+c, a+d, b+c, b+d, and c+d are represented by the mixed color bundles, and then the vectors a+b+c, a+b+d, a+c+d, and b+c+d are represented by the dark-colored fuzzy stick bundles.  In the first photograph, the white axis is pointing straight up.  Note how, at certain angles, the space is outlined by a square, and, at certain angles, it is outlined by a regular hexagon (though perspective seems to distort the shape).

I have looked on the WWW for names for the shape of the space in the 3-D variety of my coordinate system, but haven't found any.  I think it might be a new shape, but I'm not certain.