EDIT: Since this thread has basically degenerated into an interminable argument over whether I am insane in suggesting that there might be a better circle constant than Pi, an over-philosophical exploration of the minutiae of the definition of "angle", and well as a platform for promoting TGM as being already the perfect dozenal unit system, rather than actually exploring whether another similar system is even possible, I have retitled this thread so that anyone who wishes to continue the debate may do so at their leisure. It is, after all, inside the Pendlebury TGM sub-forum, which clearly was a mistake on my part. At some point, I'll start a new thread, outside this sub-forum, that will focus on actually developing the Primel metrology, where constructive comments would be appreciated. If in the process, someone wishes to criticize the choices I have made/will make in developing Primel, I'd appreciate it if they could take any quotes from the new thread, but post them here instead. Thank you.


From another thread:

m1n1f1g @ May 25 2012, 10:01 PM wrote:Maybe having a system based on something (roughly) close to an inch is a good idea. One of the problems with the TGM system is that the Maz is too massive, because a cube is big and water is dense. It just makes the system simpler if you can have a 1:1 ratio for the tangible definitions (especially if adding extra dimensions).

Yes, I concur that TGM has some problems with its volumel and massel being too large. One way to alleviate that would be to go for a smaller timel. Instead of choosing the quadciahour, which is 6 hexciadays, we could try a nice round hexciaday:

Let [P'] = alternate Pendlebury system.
Let Quantity + "-el" = unit of quantity
EDIT: Let's call this the "Primel" system. Accordingly, rather than an awkward subscript, let's just add a prime character [′] onto each quantitel name, and also capitalize the name Pendlebury-style. This distinguishes when the name refers specifically to a quantitel in this particular system, as opposed to the more generic use of the same quantitel names. E.g. "The lengthel in the Primel system is the Lengthel′". That could be pronounced "Lengthel-Prime" if we need to disambiguate, or just "Lengthel" when we're in a context where the Primel system is assumed.




So, we get a timel that's rather tiny, a little less than 2 "jiffies". (AC current in the US operates at 60_d Hz; a "jiffy" is a 60_dth of a second.) This limits the timel's use to the realm of precision science/engineering. However, the unquatimel is a very reasonable approximate third of a second. The triquatimel is a truncated "minute" of 50_d seconds, the quadquatimel a block of 10_d minutes, and the pentquatimel a so-called "duor", or double hour. A duor-based clock would be difficult to adjust to. However, perhaps we could still break up the day into 20_z hours. Each would be 60_z triquatimels long. The "minute" hand would count 6 triquatimels for each hour-number it passed, just as a minute hand on a conventional clock counts 5 minutes for each hour-number it passes. It would take some math figuring to convert hours and triquatimels into total timels, but no worse and in fact much easier than handling conventional time. And the upside is a day is a round power of the base timel, so the dozenal scale is seemless.

However, the main upside is that this reduced timel allows for other quantitels to be reduced. The lengthel turns out to be about 1/30_z of a Grafut, close to 5/16_d of an inch, or 8 millimeters. This is small but not unusable. The unqualengthel makes a very reasonable palm- or hand-sized unit, the biqualengthel a largish meter or yard. The pentqualengthel is a longish mile or nautical mile and an almost perfect double-kilometer. Even the hexqualengthel is useful; it's a close approximation for 15_d miles or 25_d kilometers.

The velocitel comes out to a close approximation of a foot per second, and, amazingly, almost exactly a kilometer per hour! Ordinary automobile speedometers with metric scales could be used as-is!

The real coup comes with the volumel and massel. At about half a milliliter and half a gram, respectively, these are small units, but definitely useful. And the triquavolumel turns out to be a very close fit for both the liter and the U.S. quart! This makes the triquamassel a close fit for a kilogram, or slightly more than 2 pounds, definitely useful. And the quadquamassel comes out very close to a quarter-hundred pounds.

All of this with strict 1:1 correspondences in all the base quantitels.

Hmm ... this might make a nice system!

Okay, so let's imagine we go with the hexciaday as a timel, making the duor a round power, but still retain the hour as an auxiliary unit. We could express times of day using the following format:



Where:

• = hours since midnight
  = unquahour, =AM, =PM
  = reading of the hour-hand
• = triquatimels and fractional triquatimels, in range ..
  = reading of the triquatimel ("minute") hand, in range ..
  = reading of the unquatimel ("second") hand, in range ..

Here's a table of times mapped to straight elapsed timels:



Edit: The hour hand would need a dial with 10_z numbered marks, as usual. The "minute" hand would need a dial with 60_z tick marks, 6 per hour mark. The "second" hand would need a dial with 100_z tick marks, 10_z per hour mark or 2 per "minute" mark. This might be tricky to achieve with a single unified dial. Perhaps the "second" hand could be placed in a small inset dial with its own marks, like you see on many conventional clocks.

Continuing deeper into mechanics:



I'm not sure how to evaluate this. The forcel, workel, and powerel come out rather small compared to the MKS system and TGM systems, but somewhat large compared to the CGS system. On the other hand, it is still possible to find an unqual power prefix that results in a unit within the desired range, with no more than one digit of exponent.

Hmm. It turns out this system shows some promise. For the sake of discussion, I want to give it a name. Rather than calling it something awkward like "Alternate Pendlebury", or "Pendlebury Prime", how about we dub it the "Primel" metrology? I've changed the name of this thread accordingly.

It's certainly a workable system. The lengthel and massel are a bit on the small side, so we'd have to accept rescaling for some practical units like with TGM, but at least the timel is a bit more philosophically defensible from a dozenal standpoint.

As we're already aware, coordinating a system of human scale units based on earth's surface gravity is a bit of a challenge. We usually experience g through more or less static forces; kinematically it's a bit zippy.

Kodegadulo @ May 28 2012, 04:44 AM wrote: Let [P'] = alternate Pendlebury system.
Let Quantity + "-el" = unit of quantity
EDIT: Let's call this the "Primel" system. Accordingly, rather than an awkward subscript, let's just add a prime character [′] onto each quantitel name, and also capitalize the name Pendlebury-style. This distinguishes when the name refers specifically to a quantitel in this particular system, as opposed to the more generic use of the same quantitel names. E.g. "The lengthel in the Primel system is the Lengthel′". That could be pronounced "Lengthel-Prime" if we need to disambiguate, or just "Lengthel" when we're in a context where the Primel system is assumed.

I've also put this into the top post.

From another thread:

Kodegadulo @ Jun 6 2012, 07:37 AM wrote:But compare this with the Primel metrology I've been investigating. There, 1 Lengthel′ = 8.2106_d millimeters, so we're talking something closer to the CGS system.  And 1 Forcel′ = 5.428_d * 10_d^-3 newtons, not as small as the dyne but much smaller than a newton. That makes 2 * 10_d^-7 newtons equivalent to 3.68470_d^-5 Forcels′. Getting rid of that factor only requires us to raise the current to 164.74_d amps (118.8X_z amps), which is eminently do-able as a Currentel′! All without violating 1:1 correspondence.

I still think losing the hour here is a big mistake. For one thing, it ruins the correspondence between the protractor and the day, unless you also completely revamp the protractor. This destroys the correspondence between hours and uncias of the radian (the Pi), which requires either (a) abandoning the time-to-protractor correspondence, or (B) adopting tau and the full circle, which is a mistake beyond our scope here.

(Arguably, you don't lose the hour; but you can only keep it by introducing this factor of 60. 60 is a nice, round number, surely, but it's weird in a context where everything else is a dozen or a power thereof. It's also a false friend; people seeing it will think sixty, not sixqua; this will cause no end of trouble. On the other hand, a TGM clock is pretty transparent in its meaning to a decimalist, even if you don't know about dozenalism.)

You've also got a very tiny length unit. It's true that you get pretty manageable lengths by taking the first power of your length unit, but the whole impetus of this system is to fix the allegedly overlarge TGM Maz and Volm. Which are you going to use more often?

This "Primel" system, if I understand correctly, was designed to fix the supposedly "too large" TGM mass and volume units, and partially the supposedly "too small" TGM time unit.

Yet here we've got a time unit that's two unciaTim; in other words, much smaller than the Tim. And in exchange for smaller mass and volume units, you get a length unit that's less than a centimeter.

Then, we get a force unit that's 5;4 pentciaMag. (That's 0;0094 newtons, of 0;0021 pounds-force, for those of us not familiar with TGM units.) Isn't that rather small?

In other words, you're trading somewhat large mass and volume units for really tiny time and length units, which cause other problems in addition to their size. Isn't this robbing Peter to pay Paul?

EDIT: Corrected mistaken statement on the Primel system's force unit.

dgoodmaniii @ Jun 6 2012, 01:13 PM wrote:Then, rather than a force unit that makes gravitational force equal to 1

But in fact I do set the Gravitel' to 1, giving me a Forcel'=Weightel' that corresponds to the Massel'.

Ah, I see it was a mistake of mine to characterize any of the Primel units as being "derived" from TGM units. Primel is not derived from TGM. It's and independently-derived metrology based on similar principles embodied in TGM, but starting from different initial assumptions. (Nor, by the same token, is it derived from SI simply because it uses some of the same choices for standardized quantities, such as gravity and the density of water.) I should simply have displayed the TGM equivalents as just that, equivalents on an equal footing with SI and customary equivalents. I've fixed the original posts accordingly.

dgoodmaniii @ Jun 6 2012, 01:13 PM wrote:I still think losing the hour here is a big mistake.  For one thing, it ruins the correspondence between the protractor and the day, unless you also completely revamp the protractor.  This destroys the correspondence between hours and uncias of the radian (the Pi), which requires either (a) abandoning the time-to-protractor correspondence, or (B) adopting tau and the full circle, which is a mistake beyond our scope here.

In fact, this system, for what it's worth, will assert a correspondence between the divisions of a day and the divisions of a circle. Those divisions will simply be dozenal, rather than binunqual. At least, among the primary units of time measure. What we do with auxiliary colloquial units like the semiunciaday (hour) can be squishier. But the squishiness and flexibility of auxiliary units has already been established as a principle of TGM (witness the binary relationships between the galvol, halvol, tumblol, and cupvol, or the enneary relationship between the cupvol and the ozvol). That principle can certainly be exploited in the Primel metrology.

And hold on here! What correspondence is there between a "protactor" and a "day"? Sure, there's a numeric correspondence between hours and unciaPis because of Pendlebury's choices of timel and angulel, but for most people that's a tenuous connection at best. A "protractor" is a visual instrument, and the corresponding visual instrument for time is the "clock". And what visual correspondence do people see between a protractor and a clock? Very little.

A Pendlebury protractor might depict a semicircle divided into a dozen semiunciacircles, or a full circle divided up into two dozen semiunciacircles. (I have argued, and will continue to argue, that that was a bad idea, because of tau and the usefulness of correlating how to cut a circle into halves, thirds, quarters, sixths, and eights, and dozenths, with how those fractions are expressed as dozenal numbers: 0.6, 0.4, 0.3, 0.2, 0.16, 0.1.)

A Pendlebury clock on the other hand, typically shows a full circle divided into unciacircles (usually numbered) and biciacircles (usually just a tick mark). However, each of these angular units, the full circle, the unciacircles, and the biciacircles, are equated to corresponding semi-subunits of the day: respectively, the semiday, the semiunciaday, and the semibiciaday. (Assuming we limit ourselves to what the short hand is measuring.) That's because TGM (and traditional) clocks are based not on the day, but on the semiday.

Do these two depictions really correlate for people? I think not. To really get a visual correspondence between hours and unciaPis, you would have to have a clock displaying the full day on a circle, with two dozen divisions. Which is exactly what the left-hand dial of my Uncial Clock Deluxe (see other thread) does, if you look at the pink numbers. But that kind of display is admittedly quite alien to people, because they've lived all their lives driven on hour and semiday based clocks.

Let's face it. To say that the day is a fundamental reality of TGM, is actually false. TGM's fundamental reality is either the semiday, or perhaps the semiunciaday (hour). The day is treated as a quite secondary thing, entering into the mix only by allowing a dispensation of binary base to intrude into an otherwise rigorously dozenal system.

With Primel, all I'm trying to do is start with the assumption that the day is a fundamental reality and a fundamental unit, and seeing what the logical consequences would be for a metrology.

As for "outside the scope", well, that's a matter of opinion. Certainly I'd concede it's outside the scope of TGM, but I'm not actually proposing modifying TGM. It's not beyond my scope, and last I checked it's a free country. All I'm trying do is to enrich our dozenal Louvre with another work of art based on similar principles, partly to help validate those principles, and partly for people to use and enjoy as they see fit.

dgoodmaniii @ Jun 6 2012, 01:13 PM wrote: (Arguably, you don't lose the hour; but you can only keep it by introducing this factor of 60.  60 is a nice, round number, surely, but it's weird in a context where everything else is a dozen or a power thereof.  It's also a false friend; people seeing it will think sixty, not sixqua; this will cause no end of trouble.  On the other hand, a TGM clock is pretty transparent in its meaning to a decimalist, even if you don't know about dozenalism.)

But, my friend, unless we institute a separate identity scheme, then every number expressed in dozenal base will be a "false friend" at first. Even 100_z! People will look at a statement like "we divide up the hour into 100_z biquaTims", and some will inevitably react, "ugh, a hundred? So, what then, we have to say two-fifty for half-past?" I don't think somehow singling out six dozen as somehow weird is particularly fair. On the other hand, if we start showing people that 6 or 60_z can mean "half", where 10_z or 100_z means "whole", then they'll start to get a better visceral understanding of dozens and their factorability. So if that means we replace the decaminute with a biciaday, and a pentaminute with a hexatriciaday, that may actually be an excellent road for getting people to think in dozens.

dgoodmaniii @ Jun 6 2012, 07:47 PM wrote:Pi is the natural measurement for a number of reasons, most of which were pretty well demonstrated in The Pi Manifesto

And all debunked quite roundly in "No, really, pi is wrong".

(I have argued, and will continue to argue, that that was a bad idea, because of tau and the usefulness of correlating how to cut a circle into halves, thirds, quarters, sixths, and eights, and dozenths, with how those fractions are expressed as dozenal numbers: 0.6, 0.4, 0.3, 0.2, 0.16, 0.1.)

Meh; when you frame the question such that it favors your viewpoint, of course it sounds better that way. Think of it this way: "Pi is better than tau because it makes it much easier to divide the semicircle. Half a semicircle is 0;6; a third is 0;4; a quarter is 0;3; and so forth." Also, your example is only dividing the circumference of the circle; when dividing its area, it's precisely the opposite circumstance (that is, an eighth of a circle's area is pi/8, while it's tau/16).

I'm afraid it's you (or rather the Pi Manifesto you are relying on) that's playing fast and loose with selective examples. Circumference vs. area is not a matter of indifference here that makes the choice arbitrary. I've already covered this in another thread but let me repeat this here for the record: Mike Hartl of the Tau Manifesto calls this whole business about the area of the circle the coup de grace that finally demolishes pi! Because in fact, area is a quadratic form (a function of the square of some variable). In such forms, factors of one half naturally arise quite independently of any other constant applied to them, be they pi or anything else, simply due the nature of performing integral calculus from a linear differential

k * x * dx

where k is any constant and x is whatever variable you please, and integrating to:

k * (1/2 x²)

If we're talking about integrating from the circumference to the area of a circle, then x = r and k = τ is the simplest and most straightforward way to go about this. Turning τ into 2π just so you can swallow that 1/2 does not yield a "thing of beauty", because there is absolutely nothing beautiful about hiding one's dirty laundry.

But we need not even invoke calculus for a resolution to this. I quote from the Tau Manifesto:

If you were still a π partisan at the beginning of this section, your head has now exploded. For we see that even in this case, where π supposedly shines, in fact there is a missing factor of 2. Indeed, the original proof by Archimedes shows not that the area of a circle is πr², but that it is equal to the area of a right triangle with base C and height r. Applying the formula for triangular area then gives

A = 1/2  b  h = 1/2 C r = 1/2  (τ  r) r = 1/2 τ r²

There is simply no avoiding that factor of a half.

Invariably, when you get beyond specific examples and examine the underlying generalities, the supposed advantages of π evaporate. For instance, imagine a circular wedge (think, a pie slice, not a π slice ) subtending an angle θ (in radians). A full circle is just the specific case of this where the angle is the full circle: θ = τ. What is the general formula for the area of such a circular wedge?

1/2 * θ * r²

There is no π or τ here, until we decide to plug those cases in. But there is that pesky 1/2.

The bottom line is that we're really measuring angles, not circles; and π is a better measurement for angles, since angles greater than π radians can always be rewritten in terms of a negative angle whose absolute value is smaller than π radians.

On the contrary we are interested in all sorts of angular magnitudes. You seem to think the only angles of interest are interior angles of polygons, which are indeed limited to less than a semicircle. But there is certainly meaning and utility in talking about angular displacements greater than a semicircle and even greater than a circle. Here's an exercise:

Exercise 1: A truck with wheels of known radius of 60_z Lengthels′ goes on a short drive. Before starting, the driver notices that the inflation valves are at the top of the arcs of the tires, so they are easily accessible, and he can easily read the adjacent embossed lettering with the recommended inflation pressure without tilting his head. Call this orientation angle 0. One of the axles is equipped with an angular velocity meter. Over the course of the trip, it measures an overall average angular velocity of 60_z RPM′ (rotations per "minute", where a "minute" = 1 triquaTimel′) The trip lasts exactly 21.306_ztriquaTimel′ "minutes".

1. How many times did the wheels rotate?

Answer:
average rotation rate: ω = 60_z Circles/triquaTimel′ = 0.06 Frequencels′
time: t = 21.306_ztriquaTimels′
rotations: θ = ωt = 1076.30_z Circles

2. How far did the truck travel?

Answer:
rotations: θ = 1076.30 Circles * (τ Radians/Circle) = ~6740.49X13_z Radians [unit conversion]
tire radius: r = 60_z Lengthels′
distance: d = θr = ~338024_z Lengthel′ = 3.38024_z pentquaLengthels′ (about 6.74_z km)

3. What orientation will the inflation valves and their labels be (how should the driver tilt his head to read them)?

Answer:
angle: θ modulo 1 = (1076.30 Circles modulo 1) = 0.30 Circles. i.e. a right angle rotation forward, so the driver will need to tilt his head from horizontal to vertical.

Now please explain to me why it would benefit us to use the Semicircle, rather than the Circle, as our unit of angular displacement? This would force us to throw in an extraneous factor of 2 for no reason. Only to be cancelled out in the end by using τ/2 to convert to Radians and from that get the distance. And to determine the final orientation, we'd have to do a modulo 2 to get an angle in the range 0.0 .. 1.E_z, just so we can express this in the range of 00 .. 1E.E "unciaPis". Why do we need this added complexity, when we could just do a modulo 1 to get the angle as a fraction of a Circle? Isn't using 1 as the modulus the ultimate vindication of the principle of doing as many things 1:1 as possible?

Rotations-per-time are clearly appropriate units for rotational rate. Rotations (i.e. full Circles) are clearly appropriate units for counting spins of a wheel. Rotations less than a full Circle can clearly be expressed as fractions of a Circle and be understood without causing the human brain to implode. Why do we need a separate and distinct unit, that incorporates an unnecessary factor 2, just for angles less than a full circle, when such angles are precisely the same dimensionality as whole rotations?

And it's been pointed out many times that pi radians is the maximum angle, the angle represented by a straight line, and the unit arc which bounds such a straight line, and so forth; all of this I consider extremely important, as well.

In fact, I recall making that argument myself. But it was really a desperately lame attempt to find some -- any -- justification for claiming that a "semi-something" was somehow of more primary importance than the actual "something". I remember what a tortured strain it was to come up with some way of naming this thing without that a pesky "semi" or "half" in there. In fact, I think I was the one who tried to coin the terms "Reversal" or "UTurn" or even "Unit Arc" for the semicircle. But I repudiate all that now.

The fact of the matter is there seems to be no natural way to reframe the description so that a "semicircle" becomes a "something-else", to allow a "circle" to become "twice something-else". Except by invoking this abstract constant "π". You might as well call a semicircle a "blah", just for the sake of calling a circle a "double-blah".

I have absolutely no problem describing the maximum possible interior angle of a polygon, the limiting case which renders the two sides into a straight line, as a "semicircle" or a "half turn", and giving that the numerical value 1/2 (positive or negative). Or +/-0.6_z. I have no problem calling a right angle a "quarter turn" and giving it the numeric value 0.3_z. What's the problem with that? What can you not calculate with the angles defined in those terms?

(And which incorporate both the circumference (in the ratio definition) and the radius (in the unit arc definition) simultaneously.)

We keep going round and round on this. Tau is a conversion factor that turns a quantity expressed in circles into a quantity expressed in radians. Pi is half that quantity. You can use that to convert into radians, but only if you start with a quantity expressed in Semicircles. How would it make it clearer to use a conversion factor as the name of unit? I want to use the Circle as the name of a fundamental unit. I don't think it would be appropriate to call it a Tau any more than I think it appropriate to call a Semicircle a Pi. It's as silly as it would be to name a Circle a Three-Sixty or a Semicircle a One-Eighty. Or an Inch a Twenty-Five-Point-Four.

By "beyond our scope here" I meant that the (crazy, imo) idea of ditching pi for 2pi is beyond our scope, not that new metric systems were beyond the scope of this forum; quite the contrary, it's precisely the scope of this forum.

Angular measure is a form of measure. It is absolutely legitimate, when proposing a metrology, to include angular measure as being within the scope of the discussion. The sizes of the units chosen is entirely a question of utility, particularly if 1:1 correspondences are important. What the best conversion factors need to be in order to achieve those 1:1 correspondences is entirely a matter of utility. Appeals to the mythical importance of a particular number, just because some ancient authority chose to give it a name, amounts to little more than mysticism and numerology.

dgoodmaniii @ Jun 6 2012, 01:13 PM wrote: This "Primel" system, if I understand correctly, was designed to fix the supposedly "too large" TGM mass and volume units, and partially the supposedly "too small" TGM time unit.

I admit it was the initial motivation for kicking this off, but since then I think we've handily dismissed the notion that a metrology must absolutely be based strictly on human-scale units and provide complete convenience via its base units, for it to be acceptable. Particularly if this is not feasible because it is trying to adhere to valuable principles such as 1:1 correspondence. If the metrology includes some derived or auxiliary units that are "human-scale" and that are still "near-by" the base units (i.e., within a small number of derivation steps, powers preferably, simple ratios secondarily), then that is actually sufficient. I think that's as fair game for TGM as I would like it to be for Primel.

We can to a certain extent make up for a base unit being "too small" or "too large", by using derived units, so long as those derivations aren't too "overboard". But a bigger question is how everything hangs together, or whether you find yourself being forced to introduce extraneous factors in order to connect from the core mechanics to other branches of science such as electromagnetism.

Yes, many of the Primel units are small. But I actually see some advantages to that. Which I think will get revealed as I dig deeper into it. It has more of the feel of the CGS system,which was a viable system at one point. And many of the Primel units are within a triqua or so of some very useful sizes indeed, with the base sizes standing in the place of what otherwise would be tricias or so.

Yes, I deliberately picked a tiny timel. It was about the only choice other than the Tim that could adhere to similar principles and not create worse difficulties than the Tim. I should point out that the Tim itself is already quite tiny if not perhaps over the line as it is. (Some epileptics might find the nearly 6 Hz rate of ticking a trigger for seizures.) So what harm in going a little bit tinier, if it can help achieve better results while still upholding the principles? You yourself have argued that the biquaTim, aka the "pentaminute" or "block", is really the first generally anthropomorphically useful time unit, and yet you can still do useful science with the Tim, and the TGM system all hangs together for the most part. I don't dispute that. But I think that could be an equally valid claim for the Timel′ and the Primel system as a whole. Let's just see how it all turns out, shall we?

Gah. I knew this was a mistake. Arguing that pi is wrong is like arguing that Greenwich is a terrible place for a prime meridian. Even if it's true, it's pointless; we gain absolutely nothing by changing it.

dgoodmaniii @ Jun 6 2012, 07:47 PM wrote:Pi is the natural measurement for a number of reasons, most of which were pretty well demonstrated in The Pi Manifesto

And all debunked quite roundly in "No, really, pi is wrong".

Bah. The Tau Manifesto is impressive only in the sense that The Star Trek Encyclopedia is impressive; it's incredible that someone would spend so much time trying to solve what's not a problem.

I'm afraid it's you (or rather the Pi Manifesto you are relying on) that's playing fast and loose with selective examples. Circumference vs. area is not a matter of indifference here that makes the choice arbitrary. I've already covered this in another thread but let me repeat this here for the record: Mike Hartl of the Tau Manifesto calls this whole business about the area of the circle the coup de grace that finally demolishes pi! Because in fact, area is a quadratic form (a function of the square of some variable).

No, this is just the tauists arguing that the pi really should be written as tau/2, which only makes sense if you're already a tau partisan. The formula for the area of a circle is .

A = 1/2  b  h = 1/2 C r = 1/2  (τ  r) r = 1/2 τ r²

There is simply no avoiding that factor of a half.

Except that you skipped the last step: Really, tauists are fussing about multiplying by two in some circumstances when all they're gaining is the necessity of dividing by two in other circumstances. Not really an advantage.

Just because there's a factor of a half at some point in a formula's derivation doesn't mean that that formula needs to keep it forever.

For instance, imagine a circular wedge (think, a pie slice, not a π slice ) subtending an angle θ (in radians). A full circle is just the specific case of this where the angle is the full circle: θ = τ. What is the general formula for the area of such a circular wedge? 

1/2 * θ * r²

There is no π or τ here, until we decide to plug those cases in. But there is that pesky 1/2.

Meh. There's no question that lots of formulas have in them. But lots of them have just pi in them, too. Some have 3pi, some have 4pi. That's not really the issue. The issue is what makes it easiest to do angle work. Pi is the answer to that question.

The bottom line is that we're really measuring angles, not circles; and π is a better measurement for angles, since angles greater than π radians can always be rewritten in terms of a negative angle whose absolute value is smaller than π radians.

On the contrary we are interested in all sorts of angular magnitudes. You seem to think the only angles of interest are interior angles of polygons, which are indeed limited to less than a semicircle.

No, not that these are the only angles of interest; just that, in the vast majority of cases, those are the ones that we're interested in. Your example is interesting, but contrived; nobody's going to choose their route by what distance will ensure that their wheels' text are pointing upward. And it can be done identically with pi, just multiplying by two. On the other hand, these "interior polygon angles" are pretty important, and using pi to measure them leads to some really convenient facts:

• * With pi, opposite angles differ by exactly 1; that is, a 0;3 unciaPi angle has an opposite angle of 1;3 unciaPi. A 0;3 unciaTau angle would have an opposite angle of 1;3 / 2, or 0;9.
  * Supplementary angles---those angles which, added together with a given angle, will equal a straight line---are simply 1;0 Pi minus the angle. In Tau terms, this is even worse than opposite angles; It is 0;6 Pi minus the angle. This makes trigonometry a bit less fun, too, because the trigonometric functions of supplementary angles are the same.
  *Trigonometry is built on the triangle; the sum of the angles of a triangle is one Pi (that is, pi radians). The sum of the angles of a triangle in tauist-world, however, is 0;6 Tau, making trigonometry less fun.
  * UnciaPis match up with the sidereal hours of right ascension (any astronomers around?) and with the time zones. Taus don't.
  </li>

Does working with tau make these cases impossible? No. More difficult? A little, but not much. But there it is.

I remember what a tortured strain it was to come up with some way of naming this thing without that a pesky "semi" or "half" in there. In fact, I think I was the one who tried to coin the terms "Reversal" or "UTurn" or even "Unit Arc" for the semicircle.

Really? How about "a straight line"? Because that's the angle we're measuring with pi: a straight line. And a straight line is surely more fundamental than a circle.

What can you not calculate with the angles defined in those terms?

Nothing, of course; this isn't a matter of necessity. But you're asking the wrong question; the status quo doesn't have the burden of proof, the affirmative does. The correct question is, "What can you not calculate with the angles defined in current terms? What compelling reason makes pi so inferior that we need to go through the trouble of replacing it?" The answer to the first is "nothing" and to the second is "there isn't one."

How would it make it clearer to use a conversion factor as the name of unit? I want to use the Circle as the name of a fundamental unit. I don't think it would be appropriate to call it a Tau any more than I think it appropriate to call a Semicircle a Pi. It's as silly as it would be to name a Circle a Three-Sixty or a Semicircle a One-Eighty. Or an Inch a Twenty-Five-Point-Four.

It's convenient to deal directly with radians without having to bandy about an irrational number the whole time. The TGM unit "Pi" makes that possible; indeed, easy.

Appeals to the mythical importance of a particular number, just because some ancient authority chose to give it a name, amounts to little more&nbsp; than mysticism and numerology.

That would be relevant to us here if (a) an ancient authority had named pi (none did; the name was only applied to it in the eighteenth century), or ( b ) anybody had made an appeal to its importance on that basis (nobody did).

You yourself have argued that the biquaTim, aka the "pentaminute" or "block", is really the first generally anthropomorphically useful time unit, and yet you can still do useful science with the Tim, and the TGM system all hangs together for the most part. I don't dispute that. But I think that could be an equally valid claim for the Timel&#8242; and the Primel system as a whole.

My only point there is that you're solving a problem by making it worse. If you're no longer trying to solve that problem, then no worries. (Though I still think TGM is better, for all the non-size-related reasons I've mentioned, plus probably others I haven't formulated here.)

Rotations-per-time are clearly appropriate units for rotational rate. Rotations (i.e. full Circles) are clearly appropriate units for counting spins of a wheel. Rotations less than a full Circle can clearly be expressed as fractions of a Circle and be understood without causing the human brain to implode. Why do we need a separate and distinct unit, that incorporates an unnecessary factor 2, just for angles less than a full circle, when such angles are precisely the same dimensionality as whole rotations?

Sounds like a special case which might call for a special unit. But it's worth noting, too, that rotations greater than a full arc can clearly be expressed as multiples of a full arc and be understood without causing the human brain to explode.

And that's really the summary of the whole pi v. tau debate, isn't it? Let's assume that everything in the Tau Manifesto is absolutely true, and everything in the Pi Manifesto is completely false. What's the benefit of using tau?

Well, we have to retrain ourselves and everybody else, abandoning a system of thinking about the circle that's subsisted for over two thousand years, so that we can avoid multiplying by two occasionally.

Not worth it. Not even close.

What if Sir Isaac Newton had formulated a law of Halvity instead of Gravity? The quantity h would feature prominently, and quite beautifully, in the equation for displacement of a falling object due to Earth's halvity:

x = h t²

The halvity of the Earth would be determined to be

h = 4.903325 meters/second²

In that case, Pendlebury might have picked a unit called the Hee:

1 Hee = 4.905025 meters/second²

(adjusted for the polar diameter of Earth).

Of course, differentiating displacement to get velocity due to halvity would give us:

v = 2 h t

Meh. So there's a factor of 2 there, big deal. But the displacement equation is just so darned elegant.

Then some upstarts come along and suggest we should treat 2h as its own quantity. They call it "gravity" and give it the symbol g, with the value:

g = 9.80665 meters/second².

Now the equation for velocity due to "gravity" becomes quite elegant:

v = g t

Ah, but when we integrate this to get displacement, a pesky factor of 1/2 intrudes:

x = 1/2 g t²

The "gravitists" try to explain that this 1/2 factor is just a natural consequence of quadratic forms and calculus. They point out that "gravity" is just a special case of a more general idea called "acceleration", where the equation for displacement is:

x = 1/2 a t²

Just plug in a = g and you get the gravity equation.

But the "halvitists" insist that the "gravitists" have skipped a step, and that we really should stick with tradition:

x = h t²

They claim that this is a clear win for halvity, and accuse the gravitists of picking and choosing their examples to favor their view. They question why, with all things being equal, should we go to all the trouble of changing this now. Their refrain is adamant: "This is crazy talk. Displacement is haitch tee squared. End of story."

--

Really, this debate is getting way too serious.

By the way, aren't we, just by being dozenalists, and just by being here, in this forum, already aboard the Starship Enterprise? Do we really hold out hope that any of this will catch on in the wider world? To the point where we actually feel we have something "at stake" that could be "threatened" by our members exploring alternative ideas and not showing "solidarity" to some kind of common effort?

I find it's actually kind of liberating not to hold out hope. That way, what've I got to lose? Why not dabble in something daring? It's just us here, you know.

---
Edit: Believe it or not, I'm not actually seeking to "supplant" TGM, any more than I think gingerbill is in proposing Idus, or Dan in working on Mesures Usuelles. If you've noticed, I've actually been making suggestions trying to strengthen TGM even while pursuing this thread simultaneously. TGM is a beautiful system, and I can clearly see the trade-offs that Pendlebury made. It's still annoying to me that he used the name of a number to name a unit of measure, but that's just a nit, really. But I want to go through the exercise I'm going through here, if for no other reason than to show that the same principles could yield another example of a coherent system, just with different trade-offs, and show what that system would be like, fully worked out. My intuition is that the exercise will reveal some useful ideas. If indeed Pendelbury made the best trade-offs then that should be self evident to people once they play with both systems.

Yes, my truck example was contrived. But I think there are applications for doing something exactly analogous to that, particularly in electromagnetics, where we're dealing with cyclic phenomena, rotations in the complex plane, waveforms and wavelengths, and difference of phase (angles in the complex plane). Where unifying the concepts of rotation, frequency, and phase angle, using the circle or cycle as the basic unit, would be an advantage. I just don't have a well-worked-out example at the moment.

I don't like being told I can't use a tool that I find useful. In fact, I feel that way about Semicircles and unciaSemicircles, as much as I do about Circles and unciaCircles. For the applications dgiii highlights, I can see the advantages of the former. If there's a way I could exploit both without losing either, I'd go for it. After all, it's as simple as:

&#964; = 2&#960;

and

&#960; = 1/2 &#964;

Can't we use whichever we want, whenever it suits us best?

dgoodmaniii @ Jun 7 2012, 02:13 PM wrote: Trigonometry is built on the triangle; the sum of the angles of a triangle is one Pi (that is, pi radians).&nbsp; The sum of the angles of a triangle in tauist-world, however, is 0;6 Tau, making trigonometry less fun.

I dunno, by my recollection, &#960; made trigonometry very painful and confusing. Thinking of a full circle as 2&#960;, a half-circle as &#960;, a quarter circle as &#960;/2 and so on, made it very abstruse, and obstructed my ability to tap into visceral intuition to understand what was going on. I know if I had &#964; at my disposal, it would have been much less painful.

From A "Tau-stimonial":

Hiya, Dr. Hartl.

I’ve been a fan of The Tau Manifesto for quite a while, I ordered the Tau Day t-shirt, etc. And today I come to you with a &#964; success story. I hope it warms the cockles of your heart as it did mine.

I’m an undergraduate student, and I spent the just-recent spring break with my family. During my stay, my 15-year-old sister begged me for help on an upcoming trigonometry test, which was based primarily on converting between degrees and radians and finding sines and cosines of all manner of wacky angles. She gave me a bundle of homeworks (all with fairly depressing scores) that showed the material covered.

I started off trying to reinforce some of the concepts she seemed to be confused about, but she said I mostly ended up echoing her teacher and that she still just “wasn’t getting it.” So I said to myself, “screw this, I’m going to do this the Correct Way,” and I started teaching her all of the same material from scratch, but using &#964; instead of &#960;. Just as I was drawing the fractional segments of a circle labeled with &#964;, &#964;/2, &#964;/3, etc., her eyes lit up—she grokked it. And when I drew the first right triangle to talk about sines and cosines, she took the pencil from me and finished explaining the material herself. And then she went back and annihilated all the old homework problems she missed.

Her strategy for the test was to do every problem with the &#964;-circle, and then to sweep over it at the end and convert every &#964; to 2&#960;. She was the first one in her class to finish the test, with a 100%.

So I (and my little sister) thank you, Dr. Hartl, for campaigning as avidly as you have for &#964;. It has enriched our lives and brought us much joy, and will probably stop the End of Days. Keep up the good fight; I intend to as well, where I can.

—an MIT undergraduate (who wishes to remain anonymous)

My kid is going to hit trig and calculus in a few years. There is no way that I am not going to equip him with this tool.

I found this interesting diagram at a blog site called hexnet.org. It's a dozenal tau-based unit circle, displaying the sign and cosine coordinates for all two dozen semiunciaCircle angles:



Kind of suggestive of something you might do with unciaPis in your TGM Book, eh?

Looks like Bluzzaro found this already:

Re: Dozenal tau unit circle
Posted by (Bluzarro) at 1969.1231.1900

This is great work, consider it swiped! Your math all looks right on to me. Although technically, leaving a radical in the denominator is frowned on, I see how it makes the patterns easier to see, and subsequently memorize if desired.
Inspired by your example, I have drawn (on paper) a similar graph of my own. Mine differs in that it is divided only into twelfths with solid lines and eighths in dotted lines, leaving out some of the lesser-used 2-dozenths.
I also labeled each "spoke" with its dozenal point value of tau. In other words, 30 degrees on standard decimalized chart = Tau/12 on yours = .1Tau on mine. 45 degrees = Tau/8 = .16Tau, 90 degrees = Tau/4 = .3Tau, 300 degrees = 5Tau/6 = .A Tau, 330 degrees = 11/12 = .B Tau, etc. This notation really brings home to me the advantages of both tau and dozenal in the unit circle.
As a first attempt, my chart looks like a combination of a clock and a pizza, which doesn't sound appetizing at all, but tastes like dessert to the mind. Looking at it makes everything just about the unit circle "click" for me.
So hey... thanks.

dgoodmaniii @ Jun 7 2012, 02:13 PM wrote: Really? How about "a straight line"? Because that's the angle we're measuring with pi: a straight line. And a straight line is surely more fundamental than a circle.

I think it would be completely confusing to say "a full circle is two straight lines". However, I just ran across an obscure term I'd long forgotten: "straight angle". So two right angles make a straight angle, two straight angles make a circle.

dgoodmaniii @ Jun 7 2012, 02:13 PM wrote:*Trigonometry is built on the triangle; the sum of the angles of a triangle is one Pi (that is, pi radians).  The sum of the angles of a triangle in tauist-world, however, is 0;6 Tau, making trigonometry less fun.

Actually, the way I'd put it is, "No matter how you measure them, the angles of a triangle add up to a semicircle."



I dunno, it doesn't seem that much harder to me to say
1 + 2 + 3 = 6, versus
2 + 4 + 6 = 10_z, versus
4 + 8 + 10_z = 20_z, versus
30 + 60 + 90 = 180.

Whether you characterize a semicircle as half (0.6_z) of a circle, or a whole (1.0_z) semicircle out of two, or two (2.0_z) quadrants out of four, or 180_d degrees out of 360_d, or &#964;/2 radians out of &#964;, or &#960; radians out of 2&#960;, it's still half a circle.

Except that half-gravity isn't a whole anything. Pi radians, on the other hand, is a whole angle, a full reversal, a straight line; as you pointed out, also known as a straight angle. It's a real thing. So the situations aren't remotely analogous.

Really, this debate is getting way too serious. smile.gif

Yes, probably so.

By the way, aren't we, just by being dozenalists, and just by being here, in this forum, already aboard the Starship Enterprise? smile.gif Do we really hold out hope that any of this will catch on in the wider world? To the point where we actually feel we have something "at stake" that could be "threatened" by our members exploring alternative ideas and not showing "solidarity" to some kind of common effort?

Hope? Yes. I don't really expect it, of course, but yes, I hope that this will happen. I think this is much more likely if we can all rally behind something concrete.

It's not that I think TGM is "threatened" by exploring alternative ideas; I've said this more than once. It's that I think TGM is just about as good as any metric system could be, and so I comment on ways in which I think other systems aren't as good. It is a discussion board, after all.

I don't like being told I can't use a tool that I find useful.

Wait, did somebody tell you that? I thought I said quite the opposite:

I wrote: Sounds like a special case which might call for a special unit.

I don't have a problem with plugging in when the formula calls for ; but I find it insane that anyone would argue for plugging in when the formula calls for . Part of the problem is that Hartl in his little manifesto comes off as a massive asshole; talking about his opponents' heads exploding and openly implying that they only like pi over tau because they're ignorant.

Rarely do pi-partisans say that nobody should ever use tau. But tauists regularly say that pi is an abomination that should be banished, and that tau is "the true constant." If all they were saying was that tau should be subbed in for 2pi, there would be no need for manifestos or arguments at all.

Can't we use whichever we want, whenever it suits us best?

Yes! So why do tauists say things like, "" and "the truth is on our side"? These are tools; we use the ones that are most convenient. And swapping out fractions of tau for every pi in every equation is not convenient, especially not in equations like the area of the circle, which is just plain easier and nicer with pi.

I honestly believe that tau is a solution hunting hard for a problem; but at least I'm not saying that tauists are perpetuating a centuries-long conspiracy to suppress the One True Constant, as Hartl does.

As for the taustimonial, I'm not impressed, largely because it's anecdotal; and it's worse than anecdotal, it's just one anecdote, and it's from somebody who won't even tell us who he is. (Anonymous anecdotes are worth even less than others, which themselves aren't worth much.) So I'll do it one better: I'll give lots of anecdotes, and you all know who I am! I never had any problems with pi, nor did I ever know anyone to complain about dealing with pi or about multiplying it by two when necessary. There. Taustimonial refuted---the refutation being worth only very slightly more than the refuted.

Now, as for trigonometry: I did certainly know of people having trouble with radians, but not because pi was in them. They had trouble with them because they had trouble figuring out what to do with that transcendental number floating around. The Pendlebury system does away with this; measure the angle in Pi, do your calculations, and at the end, all you need to do is add to it and you've got radians. If you don't want to call them Pi, don't; call them "angulels" or whatever. But the utility of the system itself seems clear to me.

I dunno, it doesn't seem that much harder to me to say
1 + 2 + 3 = 6, versus
2 + 4 + 6 = 10z, versus
4 + 8 + 10z = 20z, versus
30 + 60 + 90 = 180.

Whether you characterize a semicircle as half (0.6z) of a circle, or a whole (1.0z) semicircle out of two, or two (2.0z) quadrants out of four, or 180d degrees out of 360d, or &#964;/2 radians out of &#964;, or &#960; radians out of 2&#960;, it's still half a circle.

You're right! It's not much harder. Which is why tau is a solution looking for a problem; even where tau might be better, pi is still only very, very slightly worse. (Degrees of angle is enough harder that we can probably safely knock it out of the running, though, at least as far as new metric systems are concerned.)

And of course, I'd rephrase your last statement; "Whether you characterize a straight line as a full (1;0) straight line, or a half (0;6) of a circle, or two (2;0) right angles, or &#964;/2 radians, or &#960; radians, it's still a straight line."

I think it would be completely confusing to say "a full circle is two straight lines". However, I just ran across an obscure term I'd long forgotten: "straight angle". So two right angles make a straight angle, two straight angles make a circle.

Yes, it would; it would also be false, as two straight lines don't make a circle. But the maximum angle on two straight lines do add up to the same angle which goes all the way around a circle. And I also had long forgotten the term "straight angle," if I ever knew it, but it's descriptive and functional; I like it.





P.S. I found an interesting article that pretty strongly makes the case for being a better constant for Euler's identity (Hartl's rebuttal to the points in the Pi Manifesto by adding "+ 0" to the identity seemed weak and petulant; pi really just has the better of this one), as well as the Fourier transforms: Tau versus Pi - Proving the Obviously Untrue. A good read, abnormally so for a blog post.

That's a neat graphic! I'm busy right now with the index (indexing is a surprisingly intensive and manual job, so it's slow going; but I'm getting close to done), and I want to publish v1.1 with fixes for the problem you pointed out this week along with said index, so this will probably wait for v1.2. But it looks good.

It also points out something that I didn't notice before: corresponds a little better to the points on a unit circle than tau does.



Just a little better. :-) Really, only (-1,0), because it's x-coordinate is a simple 1, as is the coefficient on . Probably not particularly helpful to anyone. But it's funny.

What might be more helpful is that a full turn isn't or ; it's 0. (That is, it's all three; but the simplest expression for it is 0.)

Come to think of it, one full turn is really a composite angle; it's four right angles. Maybe is a better constant. I think I'll write a manifesto. :-)

dgoodmaniii @ Jun 8 2012, 01:19 PM wrote:It also points out something that I didn't notice before:  corresponds a little better to the points on a unit circle than tau does.



Just a little better.  :-)  Really, only (-1,0), because it's x-coordinate is a simple 1, as is the coefficient on .  Probably not particularly helpful to anyone.  But it's funny.

Honesty, I don't get what you mean by "corresponds". To my eye, the tau numbers "correspond better" because they relate to how far around the circle you need to go to get to those points: a quarter way round, half way round, three-quarters of the way round, the whole way round (or just stay put).

But my beef, like yours, is with the radian as a unit, and the need to grapple with a transcendental number, be it &#964; or &#960; when thinking about angles. This should only become necessary when you need to switch dimensionality from angular measure to linear measure. This is why in my last diagram, I coined some unit symbols that characterize angles visually, not in terms of abstruse numerical abstractions:

&#9689; = 1 Circle
&#9690; = 1 Semicircle
&#9684; = 1 Quadrant

In which case your table comes out as:


If we're talking intuitive understanding of angle, I think the Circle numbers win out. If we're talking avoiding dealing with fractions as a criterion for ease of use, the Quadrant numbers win out (for these particular angles). The Semicircle numbers don't impress me much.

In fact, the Quadrants appear to "correspond" better with these points, because they're all points where one coordinate is 0 and the other is +/-1.

My beef with Pendlebury is not so much that he picked the Semicircle as a unit, but that he chose to call it a Pi, beating people over the head with a transcendental number. Yes, we need to deal with &#960; at some point, depending on the application. But every time we talk about angles? Really?

An airline pilot flying south-by-southwest is going to give his heading to the regional air controller as "one dit one six Pi"? (By the way, you ought to include a graphic of a TGM compass rose.)

I've said repeatedly, I wouldn't want to call an angle unit a "Tau". But at least when people see a &#964;, they could choose to read it as a "turn", and not worry about the value 6.34..._z. But you can't look at "Pi" and not be taken all the way back to the horrors of trig class.

Come to think of it, one full turn is really a composite angle; it's four right angles.  Maybe is a better constant.  I think I'll write a manifesto.  :-)

You know, all I did was wonder out loud whether "tau" would be helpful, and all of sudden you were down my throat as a "Tau-ist". I honestly didn't know anything about the "controversy" but once I saw the manifesto I was intrigued by the reasoning. I quoted an anecdote because it rang true for me, for my experience. I don't think I'm alone.

And though you joke about /4, yeah sure, there could be some utility in using Quadrants as a unit of measure. Sometimes. If Semicircles make it easier to get supplementary angles, Quadrants make it easier to get complementary angles, and not much harder to get supplements. After all, we're dealing with right triangles in trigonometry, so numerically you could ignore the right angle and just focus on your complements. But I know that innocent suggestion is just going to spark off another debate ...

dgoodmaniii @ Jun 8 2012, 01:19 PM wrote: What might be more helpful is that a full turn isn't or ; it's 0.&nbsp; (That is, it's all three; but the simplest expression for it is 0.)

You're climbing up a spiral staircase. A dozen stories. You do a dozen full turns. But no, you haven't gone anywhere.

You're a whirling dervish, spinning at a rate of 0.2_z Freq. You do 2000_z full turns in one hour, and stop. Your head is spinning. But no, you haven't done anything.