measurement | RobertLovesPi.net

When I am asked for my height, anywhere — especially at school — I answer the question honestly. I am 1.80 meters tall.

I also live in the USA, one of only three remaining countries (the other two holdouts are Liberia and Myanmar) which have stubbornly refused to adopt the metric system. However, I am every bit as stubborn as other Americans, but, on this issue, I choose to be stubborn in the opposite direction.

It should surprise no one who knows me well that my classroom, whether I am teaching science or mathematics, is, by design, an all-metric zone. After all, like >99% of people, I have ten fingers (assuming thumbs are counted as fingers), ten toes, and almost always use the familiar base-ten number system when counting, measuring, doing arithmetic, or doing actual mathematics. (Doing arithmetic is not the same thing as doing real mathematics, any more than spelling is equivalent to writing.) Using the metric system is consistent with these facts, and using other units is not.

Admittedly, I do sometimes carry this to an extreme, but I do so to make a point. Metric units are simply better than non-metric units. Why should anyone need to memorize the fact that there are 5,280 feet in one mile? It actually embarrasses me that I have that particular conversion-factor memorized. By “extreme,” I mean that I have been known to paint the non-metric side of meter sticks black, simply to make it impossible for students in my classes to confuse inches and centimeters, and prevent them from measuring anything with the incorrect units.

To those who object that American students need to understand non-metric units, I simply point out that there are plenty of other teachers who take care of that. This is, after all, the truth.

Often, after giving my height as 1.80 meters, I am asked to give it in other units. Unless the person asking is a police officer (in, say, a traffic-stop situation), however, I simply refuse to answer with non-metric units. What do I say, instead? “I’m also 180 centimeters tall. Would you like to know my height in kilometers?”

If pressed on this subject in class — and it comes up, because we do lab exercises where the height of people must be measured — I will go exactly this far: I am willing to tell a curious student that there are 2.54 centimeters in an inch, 12 inches in a foot, and 3.28 feet in a meter. Also, I’m willing to loan calculators to students. Beyond that, if a student of mine really wants to know my height in non-metric units, he or she simply has to solve the problem for themselves — something which has not yet happened. I do not wish to tell anyone my height in feet and inches, for I do not enjoy headaches, and uttering my height, in those units I despise, would certainly give me one. Also, obviously, you won’t find my height, expressed in non-metric units, on my blog, unless someone else leaves it here, in a comment — and I am definitely not asking anyone to do that.

I might, just for fun, at some point, determine my height in cubits. For all I know, a person’s height, measured with their own cubits, might be a near-constant. That would be an interesting thing to investigate, and my students, now that I’ve thought about the question, might find themselves investigating this very issue, next week. The variability of cubits, from one person to another, makes them at least somewhat interesting. It also makes cubits almost completely useless, which explains why they haven’t been used since biblical times, but that’s not the point. One can still learn things while investigating something which is useless, if one is sufficiently clever about it.

Feet and inches, however, are not interesting — at all. They are obsolete, just as cubits are, and they are also . . . offensive. It is not a good thing to insult one’s own brain.

A Proposed New Unit for Angle Measure: the Euclid

Some angles are constructible, in the Euclidean sense that they may be constructed with the traditional geometricians’ construction-tools: a compass, and an unmarked straightedge. Examples include every angle shown above, such as the 108° interior angles of the purple regular pentagon, or the 60° angles of the yellow triangle. Angle LEN is constructible as well, and measures 48° — but to construct it, one must use compass-and-straightedge subtraction (the 108° pentagon angle HEK, minus the 60° triangle angle KEL). After constructing this 48° angle, I bisected it repeatedly, to show that angles measuring 24, 12, 6, and 3, and 1.5 degrees may be constructed as well. The 1.5° angle NET is shown with a blue interior.

Many other angles are non-constructible. For example, the angle between two adjacent radii of a regular enneagon (also called a nonagon) measures 40°, and so, because it has been proven that the regular enneagon cannot be constructed with the traditional Euclidean tools, it follows that 40° angles are non-constructible. If they were constructible, however, the subtraction-trick I used earlier to construct a 48° angle could be used, again, to construct an 8° angle (48° – 40°) — so 8° angles, therefore, are also non-constructible. Since repeatedly bisecting an 8° angle would yield angles measuring 4, 2, 1, 0.5, o.25, etc. degrees, all of these angle-measures are for non-constructible angles.

With the one degree angle on the non-constructible list, that throws into question the practice of using degrees to measure angles. As for other established units of angle measure, they have the same problem. It is not possible to construct an angle measuring one radian — nor one gradian, either. (Gradians are little-known angle-measuring units; a right angle measures 100 gradians.)

If an angle-measurement system is to be based on units which correspond to the measure of constructible angles, the blue angle above, measuring 1.5°, is ideal . . . and I am, therefore, using this angle as the definition for a new unit of angle measure: the euclid. If an angle measures a whole number of euclids, it is constructible, and this cannot be said for the degree, radian, nor gradian. (By the way, leaving “euclid” uncapitalized, in this context, is deliberate, for I am using it as a unit. This follows the convention set by other units named after people. For example, “Newton” refers to Sir Isaac Newton, but “newton” refers to a unit of force.)

One full rotation would be a rotation of 240 euclids. A right angle is one-fourth of that, or sixty euclids. The interior angles of equilateral triangles measure forty euclids, and the interior regular-pentagon-angle of 108° becomes 72 euclids, in this new, proposed system.

360 has been used as the basis of the degree for reasons both historical and mathematical. Sixty, and its multiple 360, appear as important numbers in several ancient cultures, and 360 also has many whole-number divisors, having a prime-number factorization of (2)(2)(2)(3)(3)(5).

However, 240 has similar properties. As I have shown, it is based on the Euclidean construction-rules from ancient Greece. The number 240 also has many whole-number divisors, since its prime-number factorization is (2)(2)(2)(2)(3)(5).

Just in case this catches on, I have created a symbol for the euclid, to be used in superscript form, as the degree symbol is used:

A simple “e,” by itself, would not do, for that would cause confusion with the important number e — the base of natural logarithms, among other things. That is why I included a circle, surrounding the letter “e,” for this symbol. In superscript form, this symbol for the euclid would resemble the well-known copyright symbol — but, fortunately, the copyright symbol is not, itself, copyrighted.