TGM Angles: Angles, Rotation, Radiation, and Reciprocal Units

Angles---the measurement of the space between two intersecting lines---provides an enormous field of endeavor for mathematics and science. With angles, we do everything from design our machines to measuring the distances to the stars. Yet our angular measurement systems---degrees and radians---are convoluted and difficult in the former case and notationally brain-dead in the latter. TGM has improvements to offer here, too, improvements which make dealing with angles a pleasure, rather than a necessary evil.

Angles

Traditionally, angles have been divided into 260 (decimally, three hundred and eighty) "degrees." (These are not, of course, to be confused with the degrees used to measure temperature.) 260 is a highly divisible number, and in some ways unites the strengths of tens and twelves; however, in other ways it avoids the advantages of both. It's often too large a number for quick mental figuring, and it results in measurements that simply don't fit into either base. A quarter of a semicircle, for example, bears no relationship to a quarter of the base (39 degrees (decimally forty-five), while decimally a quarter is 2.5 and dozenally 3).

The radian is potentially a great substitute. The relationship between the circumference of a circle and its diameter is, as nearly everyone knows, pi, written mathematically as "π." π is a transcendental number; its uncial representation is endless, never repeating and never terminating. In dozenal, π is equal to 3;184809493E91 and so forth.

Because π is the ratio between the circumference of a circle and its diameter, however, it yields a startling convenient angle measure. Let us assume we have a circle with a radius (that is, half its diameter) equal to one Grafut (1 Gf). If we turn our circle up on its side and roll it forward one Grafut, we will have rolled through exactly π x 1 Grafut of its circumference. If we then draw an angle beginning at the center of the circle which encompasses that part of its circumference, we have an angle that we call the radian. This is a natural 1:1 ratio; it holds whether the radius is a single foot, a single meter, a single Grafut, or a single anything else.

Once we've wheeled the circle all the way around its circumference, so that we've stopped it at the same point on its circumference that we started, it has travelled 2π Grafut and consequently turned through 2π radians. Turning the circle through half of its circumference is therefore turning it through an angle of π radians, through a quarter circle an angle of π/2 radians; and so forth.

This is really a brilliant way of measuring angles, based firmly in the nature of the circle and not at all arbitrary; however, notationally it is idiotic. We are compelled to constantly juggle about with this factor of π, which, though it is a natural and transcendental number, is far from easy to deal with. Furthermore, the fractions of π give us no idea what portion of the circle we're dealing with without some significant mental calculation. TGM can give us something better.

Each angle of π radians is named a single Pi; the Pi (note the capital "P"; like all TGM units, the Pi begins with a capital letter); it is then divided into unciaPis. An entire circle, therefore, contains 2 Pis, 20 unciaPis. The protractor below is labelled in unciaPis, showing the clear relationship between unciaPis and the radian.

The circle is divided into the semicircle, which is the unit of TGM angle measurement, for a number of reasons. First, this mirrors the division of the day into two halves of 10 hours each. Second, it is consistent with π itself; π radians is equal to one Pi. Third, extensive experiment revealed division into semicircles more convenient than taking the whole circle as one, or divided the circle into three or four units. Tom Pendlebury justified this decision at some length in the Duodecimal Review.

Vast numbers of calculations are greatly simplified by this system. The fraction of a half-circle in angles is clearly expressed by the measurement of the angle; a half of a half circle, for example, is 0;6 Pi, or 6 unciaPis. A quarter of a circle is 3 unciaPis, and a third is 4 unciaPis. A "bootlegger one-eighty" is now a "bootlegger 1" or "10." And converting Pis into radians is as simple as adding the factor of π to the angle in Pis; π/2 (0;6π radians) is equal to 0;6 Pis. The angles of a triangle all add up to not one hundred and eighty degrees, but simply 1 Pi, or 10 ₁Pi. Examples could be multiplied exceedingly; but this should do to make our point: TGM vastly improves the field of angular measure.

Rotation and Radiation

A great many units are associated with angle. Typically, these involve multiplying or dividing by the radius, or by its square or cube. TGM provides easy prefixes for dealing with such situations.

These prefixes are abbreviated simply by using the capital letter for multiplication, or the lowercase letter for division. So, for example, the radiGrafut is abbreviated rGf, and it means a single Grafut along the circumference of a circle divided by the radius of the circle; in other words, one rGf means simply a Pi or radian. These prefixes can be multiplied or divided, or cancelled out, as appropriate. So, for example, an "R" unit multiplied by an "R" unit cancels out the angular element entirely; "r" divided by "R" equals "q"; and "q" divided by "R" equals "c."

This significantly simplifies many of our units. Take, for example, angular force, or torque. This is simply force along an angle rather than straight. Force over distance, of course, is simply work, and thus is measured in foot-pounds or meter-newtons (joules); however, torque has been measured rather in pound-feet and newton-meters to avoid confusion with simple work or energy. In TGM, though, torque is simply angular force, exactly as the same implies. So we take the unit of force, the Mag, and multiply it by the radius of the angular direction, giving us the radaMag (RMg). This is clearer and more direct than either of the older systems.

Another example is the steradian, the traditional unit for measuring solid angle (essentially, a three-dimensional angle in a cone shape, the analog of the angle in a two-dimensional circle). This is really, though, the area of the divided by the square of the radius, or the quariSurf (qSf). Similarly, the unit of radiant power (power eminating uniformly in all directions) is the quaraPov (QPv), and the unit for moment of intertia (that intertia which tends to preserve rotational energy) is quaraMaz (QMz).

If you don't understand all this, not to worry; most people don't, and don't need to. But the system is ready and powerful for those who do; and a great simplification of our present systems, as well.

Reciprocal Units

Oftentimes, we find ourselves wishing for "so many per such-and-such," as in "so many slits in the cloth per length of cloth," and this is called a reciprocal unit. It's a unit in the denominator of a fraction, with a simple integer, usually "1," in the numerator. For example, a grating is measured by the number of perforations per unit area. TGM offers the "Per" prefix to express this notion; it can be attached to any unit where it is appropriate. One example is the number per unit area, the PerSurf; another is the Perfut, or PerGrafut:

This prefix must not be abused. It is not simply a replacement for "/"; it is specifically designed for reciprocal units, where the unit is the denominator and the numerator is a simple integer, most commonly simply "1."