Comments on Angle Units

There are three different kinds of circle, which use separate units and constructs.

1. Circles in the sky, measured in units of time and calendar.
   
2. Real circles, measured by their diameters, the value \( \pi\) measures the roll against diameter.
   
3. Arcs, which are measured by right variation at a distance.
   

Circles in the sky are measured according to the day-cycle (equatorial circles give a zenith-marker pointing to siderial noon), and the elliptic traces out the movements of the sun, the degree corresponding close to a day.  In either case, these divisions are properly dealt with with the derivation of time units.

Radian measure is usually applied to small variations over a distance, such as gradients (1:12 or 8.3% etc), or to distances at a distance (the military 'mil' is taken to be yards at 1000 yards, and \(\pi=3.2\) in NATO countries, or \(\pi=3.15\) in Sweden.  Because this system closely matches the units inferred in the Taylor series, and the radian is the angle coherent in the relation of torque = moment of force, it is used in defining systems.  However, it is not essential for geometry, and I use it mainly when i write programs.

The real-circle is one which one might visit all parts of, or hold in the hand.  It is given by its diameter, symbol Ø.  In Sir Thomas Heath's "history of greek mathematics", the sumerian angle for real circle was to divide the circumference into 180 ells, of 24 digits, the diameter taken to be 60 ells.  This is the unit that i divide into 120 parts, and so forth.  

The measures of solid angle are by area, there are three units, the steradian, the degree excess, and the square degree.

1s00 (sphere=1)

System	Plane Angle	Solid Angle
Natural	\( 2\pi\) radians	\(4\pi\) Steradians
Common	\(360\) degrees	\(720\) degrees Excess
Astronomy	\(360\) degrees	\(129600/\pi\) square degrees
Twelfty	1c00 (circle = 1)	1s00 (sphere = 1)

The ratio between the radian:degree is the same as the steradian:deg excess: sq degree.  These units suppose a surface-fraction.

The units of the twelfty system relate to the fraction of whole space occupied by a thing.  The letter c or s or g stand for the complete space in 2, 3, 4 dimensions, and replaces the radix-point.   The solid angle of a cube is s15 at the vertex (ie 1/8), but s30 on the edge (1/4),  The solid angle at the edge is equal to the dihedral or margin angle, which is c30.  The largest solid angle by any vertex of a regular solid belongs to the twelftychoron, {5,3,3} which gives g38.24.

Twelfty angle and astronomy

When one supposes a division of the circle to 12 signs of 10 degrees, and thinks consistantly of this, it becomes easier to divide the month and year into 120 parts, especially as these are in effect ¼ and 3 days respectively.  So one can guage the phase of the moon, and its position of sky, and even with these crude measures of eye-estimate, derive a time to the nearest hour.  Full moon is taken as c60, new moon as c00.  So if you see the moon is waxing, and its at c40, and at c50 in the sky, (zenith is 60, nadir is 0) then the sun is trailing it by 40, makes it c10, or 2am.

With the siderial time, one would convert the year to a fraction.  It's 31/8/19 today, but since my calendar begins on 1/3, we have 6 full months and no days.  So it's y60.  The present time of 21:13, translates to twelfty as 9.73, or d96.1.  So the current siderial time is 96.1 - 60 = 36.1 (or 7 am).

It is possible to use a common angle through-out.

wendy.krieger wrote:one can guage the phase of the moon, and its position of sky, and even with these crude measures of eye-estimate, derive a time to the nearest hour.  Full moon is taken as c60, new moon as c00.  So if you see the moon is waxing, and its at c40, and at c50 in the sky, (zenith is 60, nadir is 0) then the sun is trailing it by 40, makes it c10, or 2am.

There appear to be three explicit angles here: the altitude angle or angle of elevation of the Moon, the angle between the Moon and Sun, and the hour as an angle which one is to estimate from the other two. The implication was of the hours after midnight being approximated by the subtraction of the angle between the Moon and Sun from the polar angle against the nadir for the apparent height of the Moon. An angle could not be equal to the sum or difference of two other angles unless all three angles are in the same plane, or if there are positive and negative errors that cancel each other out through averaging over time. However, this is an instantaneous calculation without averaging. Granted, the Sun and Moon may be approximately in the same plane which is the ecliptic, ignoring the inclination of the orbit of the Moon to the ecliptic plane which increases its maximum declination above the solar. While it is possible at an instant for the ecliptic to pass over the zenith of latitudes between the tropics, this is not generally true at all times.

In any case, even if the ecliptic were at the zenith, the calculation by subtraction would not give the hour angle exactly because that would still lie in a different co-ordinate reference frame, the equatorial system, rather than the ecliptic. So, this would only work properly by assuming the ecliptic and the zenith and therefore the latitude to be in the same plane as the terrestrial equator. Otherwise, the circles of the ecliptic and equatorial are projected one onto the other, which causes contraction of angles near the nodes which are where the two planes intersect, just as a circle viewed at an inclination would appear elliptical.

But the zenith is unlikely to be in the equatorial plane, as the latitudes of most places on Earth are not at the equator. Now, what struck me immediately was the high value for the angle of altitude or elevation of the Moon, giving an angular distance from the zenith of \(\pi/6\) radians. The minimum possible angular distance from the zenith is the difference between the latitude and maximum declination, which happens when the luminary crosses the meridian of the observer. This means that the latitude of the observer could not have been closer to the poles than the latitudes of plus or minus about sixty five degrees North or South of the equator.

But assuming co-planarity of the ecliptic and equator, the latitude could not be more than plus or minus about thirty degrees from the equator because there would be no declination. The latitude of the observer cannot be more than the angular distance of the Moon to the zenith because the latter would be the hypothenuse of a right-angled spherical triangle of which the former would be another side not more than a right angle because its range is between the equator and terrestrial pole.

In that case, the cosine of the angle for the number of hours to crossing of the meridian by the Moon ignoring its independent motion would be the sine of its angle of elevation or altitude over the horizon divided by the cosine of the latitude of the observer, from which the hours of the Sun after midnight could be approximated by subtracting from a half day.

The inexactitude remains however while the three angles are in three different co-ordinate systems, and I think it would be possible to choose values for the erroneous estimate to be beyond the nearest hour.

You can even figure out where the sun and moon are on an oblique curve. This does not have to be straight.

If the indicated time was 2am, the sun would not be in the sky. The angles in question are c50, being the position in the sky, where the moon is and c40, the estimated phase of the moon, based on the shape of the crescent. If it is C-shaped, the left edge is lit, and the moon is waxing, while if it is in the right, the moon is D-shaped, and is waning.

The new moon is at c00, so 1/3 of the month has passed, and the moon is c40 ahead of the sun. But if the moon is at c50, the sun is at c10, which means that it's 60° below the horizon.

There are actually four angles involved, the fourth one denotes the position of the as-yet unrisen sun.

If you are familiar with the position of the sun in the sky at various times, you can suppose a portion of the sun-circle written in the visible sky. It moves with the seasons, this is true, but the process is much the same.

I am not rally sure what radians have to do with this.

wendy.krieger wrote:If the indicated time was 2am, the sun would not be in the sky.

By a simplified spherical model, the Sun can be in the sky at 2am during the peak of Summer at latitudes closer to the poles than plus or minus about an angle the gradient of which is two. So, the Sun cannot be in the sky at 2am for observers between latitudes about +63.4 and -63.4 degrees. Actually, the Sun would appear to rise earlier than this time during midsummer closer to the poles and would always be in the sky within the Arctic or Antarctic circles at that time of year.

wendy.krieger wrote:the estimated phase of the moon, based on the shape of the crescent.  If it is C-shaped, the left edge is lit, and the moon is waxing, while if it is in the right, the moon is D-shaped, and is waning.

This is for viewers in the Southern Hemisphere. In the Northern hemisphere, the Ɑ-shaped or C-shaped lune or decrescent is waning and approaching the Sun, while the D-shaped or Ɔ -shaped increscent is waxing and moving towards opposition.

wendy.krieger wrote:the sun is at c10, which means that it's 60° below the horizon.

The Sun can never be as low as 60° below the horizon at latitudes as close to the poles as about 53.5° North or South of the equator.

Thus, you have narrowed your location down to somewhere in the Southern hemisphere outside of Antarctica.

wendy.krieger wrote:There are actually four angles involved, the fourth one denotes the position of the as-yet unrisen sun.

It appears that the angle of elevation of the Moon above the nadir was given and not the angle of azimuth of the Moon to meridian transit. So, an azimuthal co-ordinate of the Moon was not stated, only an azimuthal difference between the Moon and Sun which counts as a single angle. If the angles are not in the same plane, at least four angles are needed to do a spherical trigonometry calculation.

wendy.krieger wrote:I am not rally sure what radians have to do with this.

In the Original Post, radians were in the table as a natural measure for plane angles. I am just so used to using radians which are required as input for trigonometric functions in software. An inconvenience of spherical trigonometry in the field is the need to calculate the trigonometric functions by calculator or tables as the Maclaurin or Taylor series takes too long to compute by hand. To use these functions with a calculator or tables, the angle must be converted to either degrees or radians. It is best to use radians but write the angle in terms of pi or tau in the base of choice.

The sun moves on generally a small circle in the sky. This replicates that most of the time it's not on the equator.

In relation to the angles of movement, these are meant to be taken on the current tracks of the sun and moon. So there is a complete circle between c00 and cE9, and some of this is hidden below the horizon. c20 is usually below the horizon, but not necessarilly 60° below the nearest point from the horizon. It is 60° on that circle, at the lowest part of the circle (which is usually, but not always below the horizon.)

The descriptions are for the southern hemisphere. I am told from people who have been to the northern hemisphere, that the sky and the moon are upside-down, and that cyclones and the sky both go backwards. But this is of some sense, if one supposes the 'spin-down' rule used in the south, is actually a 'spin-north', and the spin-vectors emerge from the ground there. Still, the translations are pretty much the same, up to mirror-image, if i am not mistaken.

I have little use for radians. I use fractions of space, either as a greek fraction, (x parts, where y make the circle), or fractions given in base 120. These units are coherent with the right prismatic product, and the angle of section is identical to the angle of space. An octagon-prism gives at its verticals, an angle of c45, and this is also the solid angle on the edge (s45). The solid angle at the vertex is s22.60 = s45 * s60. This same system is used in the higher dimensions, where I usually do stuff.

The other angle I use is the tegmic-radian, which relates to the solid radian by a factor of (n-1)! tegmic radians = 1 prismatic radian. P19r is already bigger than the sphere, while the solid angle of a simplex is in the order of 1 to \( \sqrt{n/4}\) tegmic radians.

wendy.krieger wrote:In relation to the angles of movement, these are meant to be taken on the current tracks of the sun and moon.

This angle seems to be the ecliptic turn rather than angle of altitude or elevation or compass bearing.

The way to measure the time would be to face the pole around which the stars appear to revolve. In the Northern hemisphere this would be marked by the star Polaris, but there is no really bright star at the Southern pole. Hold up a protractor facing the pole and measure the angle to the Moon from the vertical antimeridian. Use this for your angle c50. Unfortunately, the Moon tends not to be near the pole for seeing them in the same field of view, so some guesswork may be in order. Then, subtract your phase angle c40 to get the hour angle of the Sun after midnight, and convert this to time.

wendy.krieger wrote:An octagon-prism gives at its verticals, an angle of c45, and this is also the solid angle on the edge (s45).  The solid angle at the vertex is s22.60 = s45 * s60.

The angle subtended at the centre of the octagon by its adjacent vertices is \(\pi/4\) radians which is 45 degrees. The obtuse angle between edges of an octagon is \(3\pi/4\) radians. This is the same as the internal dihedral angle at the vertical edges of the right prism having a horizontal octagonal base.

The solid angle at the vertices of the octagonal prism is the sum of the three dihedral angles at the edges connected to that vertex minus pi, so this is \(3\pi/4 + \pi/2 + \pi/2 - \pi = 3\pi/4\), and the number of this solid angle at the vertex is indeed the same number as that of the dihedral angle at the vertical edges. To get this as a fraction of the total space, divide it into the total space solid angle \(4\pi\) of a sphere.

A dozen plus three years ago, I was able to derive from the areas of the lunes made by the dihedral angles this formula for the area of a spherical triangle from which the solid angle at the vertex may be calculated:
$$Area = (α + β + γ - \pi)r^2,$$
where \(α, β, γ\) are the dihedral angles, and \(r\) is the radius from the vertex or centre of the sphere.

The units of 'common solid angle', are degrees excess, precisely because spherical angles exceed the plane angles of the polygon, by a measure proportional to the area. In essence, the vertex-angle are measured by the margin-angles, which become the spheric angles on the sphere. This is a triangle, the spheric angles are c30, c30, and c45, less the plane sum of c60, gives c45 excess, or c22.60 of space.

The arcs are ths side of a spherical trianle, are largely ignored. In the case of prisms, it is equal to the opposite angle (being a double-right-angle triangle), but this is not usually the case. The tetrahedron has a spheic arc of 20 (ie 60 degrees), but the margin-angle (which becomes the spherical angle, but remains a margin-angle), is 23.60, so the excess is c10.60, and the solid angle is s05.30.

In relation to the star phases and the star tracks, the phase of the moon is derived from looking at a rotating half-lit sphere, while the tracks are as by the current sun-track, which is not a great-circle, but a circle that varies from c22.30 to c30.00 (67 to 90 deg), depending on what part of the year it is.

In the summer solstace, in december, the sun has gone as far south as it might, and the whole circle is visible from the capricorn-artic, while in the winter-solstace, in june, the sun has retreated as far north as it goes, and its full trace is seen north of the cancer-artic. In four dimensions, there is an artic circle for every sign of the zodiac.

At the equinoxes, the sun is directly over the equator, and the trace is a great circle.

wendy.krieger wrote:This is a triangle, the spheric angles are c30, c30, and c45, less the plane sum of c60, gives c45 excess, or c22.60 of space.

The notation of the double-digit angles in twelfty is similar to dozenal pergross tau radians except the units place in twelfty has ten pieces, whereas in dozenal the units place has twelve pieces. So an angle of 5 in twelfty is half of the units place, whereas in dozenal half of the units place is 6. Thus, c30 and c60 for example are the same in twelfty as in dozenal, but twelfty c45 is four dozen plus six pergross tau radians, or forzei six rather than forty-five. It is a third plus half a twelfth of a turn.

The full spherical solid angle in twelfty, if I am not mistaken, is s100.00. This puts the solid angle at a vertex of the right octagonal prism as three square quarters of the full space. This makes sense, because the plane angle at a vertex of an octagon is three eighths of a full turn. Chopping off a right angle from this would produce a solid angle filling an octant of space, while the rest would fill half an octant, or 1/2 times 1/8 which is a square quarter of the full space solid angle, and 1/8 plus 1/4^2 is three square quarters. This should be s22.60 in ratio to s100.00.

wendy.krieger wrote:The arcs are ths side of a spherical trianle, are largely ignored.

Sometime between autumn of the year 2000 and before the autumn of the year 2002, I realised that the dihedral angles of polyhedra can be calculated from their face angles and mentally derived a formula for doing so. Before then, I had been calculating the dihedral angles with ad hoc methods on a case by case basis and was expressing the calculated angles in terms of slopes. At that time, I calculated angles and lengths in the four-dimensional simplex, using almost only slopes and the theorem of Pythagoras. I used other methods for the same results later.

wendy.krieger wrote:The tetrahedron has a spheic arc of 20 (ie 60 degrees), but the margin-angle (which becomes the spherical angle, but remains a margin-angle), is 23.60, so the excess is c10.60, and the solid angle is s05.30.

The slope of the dihedral angle of a regular tetrahedron is the square root of eight, which is about twelfty c23.61. The spherical excess would then be c10.63. The solid angle at its vertex goes into the full space solid angle about 22.8 times, so it is about s05.32, the number of which is half that of the dihedral angle.

Astronomic angles

wendy.krieger wrote:the current sun-track, which is not a great-circle, but a circle that varies from c22.30 to c30.00 (67 to 90 deg), depending on what part of the year it is.

These angles are the declination of the Sun subtracted from 90 degrees. They are the geographical polar co-ordinate. The path of the Sun is a great circle on the ecliptic sphere but inclined to the equator. The Sun should be about c22.22 or c22.23 from the pole at its closest approach.

wendy.krieger wrote:In the summer solstace, in december, the sun has gone as far south as it might, and the whole circle is visible from the capricorn-artic, while in the winter-solstace, in june, the sun has retreated as far north as it goes, and its full trace is seen north of the cancer-artic.

They are called the tropics of Capricorn and Cancer because those were the astrological sun signs of the solar position during the solstices. These are tropic latitudes where the Sun is directly overhead at a solstice. The Arctic and Antarctic circle latitudes are where the Sun does not rise or set for part of the year.

The notation in twelfty is the treating the circle as unity, the division is then according to the base. So the dozenal divisions relate to the twelfty, simply by converting the angle as a fraction, eg c1340 = z14, represents a ninth part.

Like the sumerian numbers, it represents a division-base, as 13.40 would translate to 1600 d, or dozenal E14. Division-bases extend to the right, with more places of fraction.

One of the features of using the circle=1, is that the computer FRAC() function returns the proper part of the circle, without further need to divide.

With the twelfty-angle, the margin-angle of the twelftychoron {5,3,3} of 120 dodecahedral faces, is 144° or c48. The solid angle at the edge, is the same as the spherical excess of a triangle, gives 252°E, (corresponding to a prism-angle of 126°), but in twelfty c42. The vertex-angle was found before the number of radians to the glome, makes c38.24, which is very near \(\frac 1{\pi}\). It turns out that the cubical radian is c06.09.60, which makes the tegmic radian c01.01.70. The solid angle of a pentachoron must be bigger than this, is c01.20.V8.

Sky-angles

In four dimensions, the tropics form a torus, or margin equidistant from a great circle. The sun runs always over this, gradually moving through the whole range, on a helix, giving tropics for all seasons.

The artic is a similar shape, the sun being on the horizon at midnight, during the season 6 months removed, so when the sun is over the tropic of capricorn, it is horizon-skirting in the capricorn-artic. These are shifted c60 on the third coordinate, so they are 'opposite', specifically, their great arrows are orthogonal.

But the sun for always being over a tropic, it moves through the year, so you have a tropic of leo, and a leo-artic, and so forth. The three-dimensional case is like those spinner-wheels that are used in place of dice: it shows only two parts of a full circle.

Of bases generally

Bases give an alternate world to the modeller and historian. In effect, one can unshackle the current conventions, and try something different if one wants to. The thing I was trying to do with angle here, is to replace time-measure (degree=day, right-acession=hh:mm siderial), with a single twelfty-angle.

In general, a different base creates a different world, and one is not required to slavishly follow the whims of ISO31 or any other dictate, when setting ones own system up. The work i have done with the electrical systems is such that one can read CGS, HLU or SI formulae and convert them into any other set of units.

wendy.krieger wrote:With the twelfty-angle, the margin-angle of the twelftychoron {5,3,3} of 120 dodecahedral faces, is 144° or c48.  The solid angle at the edge, is the same as the spherical excess of a triangle, gives 252°E, (corresponding to a prism-angle of 126°), but in twelfty c42.

More than ten years ago, I discovered that the ditopic angle between dodecahedra folded about a four-dimensional space is equal to two fifths of a perigon. In radians this ditopic angle is \(4\pi/5\) and in degrees it is 144°.

The dihedral angle in the dodecahedron was calculated probably about the year 2001, and has a slope ratio of two to minus one. This is about 116.6 degrees, or thirzei ten zot seven six pergross tau radians.

wendy.krieger wrote:The sun runs always over this, gradually moving through the whole range, on a helix, giving tropics for all seasons. [...] But the sun for always being over a tropic, it moves through the year, so you have a tropic of leo, and a leo-artic, and so forth.  The three-dimensional case

The Sun is always directly overhead of one latitude or another. It runs between the tropics of Capricorn and Cancer and back again during the course of the year. The tropic of Leo would be just South of that of Cancer.

wendy.krieger wrote:Bases give an alternate world to the modeller and historian. [...] In general, a different base creates a different world,

The motivation with dozenal is not to create an alternative world, but rather to improve the real one. Angles measured in degrees, minutes, and seconds, and time measured in hours, minutes, and seconds are not in one base with decimal. It would be better to have one base for all, but time and angle do not use a single base and have not become decimal, which is ill-suited to division of circles. So, preferably, the base of numeration and the way time and angle are divided would be changed to dozenal, or the dozenal part of division for time and angle that already is used would be retained. When doing calculations, the one angular system of radians for all is used, but the awkward conversion of the results into hours or degrees in different bases could be avoided if the base of numeration throughout were dozenal.

It is important to understand the processes and history here.

Angles of right ascession are in hours, minutes and seconds, since the zenith-marker marks the siderial time. There is a metric time system by GrandAdmiralPetry, which divides the day into 40 demur, of 1,000 hesits. This is a direct overlay of the metric grade onto the day-circle.

If instead of inventing different angle-systems, we suppose the same circle-fraction applies to the circles of the sky (which is what I have done in twelfty), we covert times to fractions of the day, month and year, and use these angles.

Of course, it's a different system, but you won't really know it's better until you use it.

Likewise, calling the artic and antartic as artics of zodiac signs is more into relating that their functions are parts of the year too.

Some ideas work out, others do not. I had a plan to use Stevins-style columns in twelfty, using 1° as 1 = sphere, and minutes, seconds, and thirds as the first, second and third division of the circle.

One of the things that drove twelfty into the fore, was that I used it with other content that people came to view. It wasn't twelfty, but the polytopes they were looking for. I gave it a generous dose of twelfty-and-translation.

The thing with dms is that even though there is a move to dd (degrees=decimal), dms is what the vast bulk of records are preserved in, and you still need dms imput and output to make these function.

But you can easily smuggle in dozenal or whatever into daily life. For example, the directions become bearings on the clock, and thus 9 o'map is west, and 12 o'map is north.

I'm not really in favour in following the dictates of ISO 31, or the SI system. It is in effect, the wreckage of a hundred years before it was introduced. But the cost of changing from something like CGS to SI, although minimal, has taken nearly 50 years, and is by no means complete, because of inbred flaws of the SI.

When metrics were written in the eighteenth century, they had a clean slate, and could implement the thinking of the time. By 1840, a large part of that was obsolete, and the electrical units of 1860+ were already obsolete beyond 1900. But the first appearence of an SI-prototype was 1904, is a stitch-together frankenstein-monster, its seams still painfully obvious.

The way I view twelfty is as a hydrid of bases twelve and ten that has use as a supplementary base to decimal that prepares for a transition to dozenal. Numbers based on twelfty look good in both decimal and dozenal. Their advantage is that they can be more easily converted to from decimal because of how numbers in decimal have been rounded to fifths.

However, if there were a clean slate, would the twelfty system be more cumbersome than dozenal? Sure, in twelfty, fifths of a circle for example could be written more neatly, but these angles are not used in lattices of the spaces used in measurement, and thus would not be relevant for physical metrological scales which apply to all orders of magnitude.

The fivefold symmetry, like sevenfold, is viewed more as a decoration and is not structural, except perhaps in geodesic domes.

The way I use angles in astronomy is as fractions of a circle. I think of them as tau radians, but to be conventional I write them as pi radians. I often find pi radians to be a nuisance conceptually in comparison to tau radians.

For angles in geometrical figures, I use variations of slope or the trigonometric functions.

wendy.krieger wrote:There is a metric time system by GrandAdmiralPetry, which divides the day into 40 demur, of 1,000 hesits. This is a direct overlay of the metric grade onto the day-circle.

There is the gradian system, which if I recall correctly was more successful in Scandinavia, that divides the quadrant decimally.

wendy.krieger wrote:The thing with dms is that even though there is a move to dd (degrees=decimal), dms is what the vast bulk of records are preserved in, and you still need dms imput and output to make these function.

To convert the minutes and seconds to degrees, they are converted to fractions of a degree by dividing them by the first or second power of sixty. The same can be done to convert minutes and seconds to dozenal more easily than to decimal if the minutes and seconds are rounded to multiples of five. To convert to a dozenal angle, the degrees would be converted to a fraction of the circle by dividing by two-and-a-half square dozen. Degrees rounded to multiples of five would convert more neatly into dozenal than decimal.

Pentagonal quasicrystals are indeed known.

Astronomy is best served by full cycles. Classical astronomy uses time-circles, which require less conversions. But they are less reliant on angles, although thirds are useful.

Slope corresponds to tangents, I suppose. I use double-versines mostly.

Actually, when I was laying out the fast geometry, there was a need to write fractions and their differences, such as \(\frac 13-\frac 17\) sort of thing. Twelfty was not in the field, came from nowhere, and became the ideal base for this.

It's not so much the simple fractions, but handling inverse factorials that really matter. 120 does this quite nicely. The next competitors are 30, 6 and 18. 80 has a good show, but in the long run, bases like 10 and 12 just don't make the cut.

Angles in Crystals

wendy.krieger wrote:Pentagonal quasicrystals are indeed known.

The prefix "quasi" says it all. Crystal symmetries arise from the possible lattices or tessellations of space. Fivefold rotational symmetry does not participate in lattices of Euclidean area or volume. Thus, division of the circle into fifths is not needed for description of angles in crystals.

For the plane, the tiles of regular polygons are of triangles, the square, and hexagons. Their angles which divide the circle into thirds, quarters, and sixths are most important and the base for numeration and division of angle should be chosen for being capable of representing them simply. Base twelve succeeds here, while decimal does not do well.

In three dimensions, the symmetries of crystals include these angles. Thirds are important for trigonal or hexagonal crystals. Thirds of a full turn angle are also important in cubic crystals because of the threefold symmetry seen when looking at a vertex of a cube eclipsing its centre.

Trigonometric functions

wendy.krieger wrote:Slope corresponds to tangents, I suppose.  I use double-versines mostly.

There are many ways of relating ratios of lengths to angles which may be used, and they are most useful if they can be related to the common trigonometric functions of calculating utilities and tables.

wendy.krieger wrote:Actually, when I was laying out the fast geometry, there was a need to write fractions and their differences, such as \(\frac 13-\frac 17\) sort of thing.  Twelfty was not in the field, came from nowhere, and became the ideal base for this.

It's not so much the simple fractions, but handling inverse factorials that really matter.  120 does this quite nicely.  The next competitors are 30, 6 and 18.  80 has a good show, but in the long run, bases like 10 and 12 just don't make the cut.

These reciprocal factorials could have arise in terms of Maclaurin series for trigonometric functions of angles. For example, the expansion for the sine function of angle theta is
$$\sin{θ} = \sum_{n=0}^{n=\infty} (-1)^n \frac{θ^{2n+1}}{(2n+1)!}.$$
Few people would grapple with such a summation series when calculators and tables are available. There are schools where the use of trigonometric tables, except for the special angles of eighths and twelfths, is no longer taught due to the availability of calculators. The summation by hand is hardly thought of as "fast". If a summation formula were being used, the factors of the denominators would already be known.

How much would the base 210, alternating perhaps as fourteen and fifteen or ten and twenty-one, be a contender for this purpose? I think base twelve should be fairly good for eliminating prime factors quickly taking into account the frequencies of the most common prime factors two and three.

The whole process of selecting a base is to reduce both the size of the leader and period. In decimal, the fraction for 1/6 has a one-digit leader '1', and a one-digit fraction '6'.

Base 210 would certainly reduce many of the periods, the primes 11 and 19 having a single place, but the leaders would be annoying. For example, 1/16 = 0;13,26,52,105. Compare this with 60 0;03,45 or 120 0;07,60. Loosing 7 is not a major drama, and the factorials will have more trailing zeros. (60, 120, 12 give two trailing zeros per 4 factorials, while 210 requires six places).

The main source of factorials is symmetry groups. So the largest symmetry group in 8D is E8, which is 3.43.24.00.00 dec 696729600 doz 175400000. But the reciprocal is 35:85.85.H-5, where the decimal and dozenal run to many places of period.

Code:

[d:\]rxc #12 1/twe(3,43,24,0,0)
 3585:8585 8585,8585 8585,8585 8585,8585 8585,8585 85 H-6  (twelfty)
 1.435 277 42,915 931 804,820 693 709,582 598 471,487 E-9    (decimal)
 7.4 10 5,0 10 8,4 4 7,11 5 5,0 0 9,10 5 10,9 2 3,1 10 2,7 3  E-9  (dozenal)
 15.14 12 10,17 9 7,15 11 7,9 2 13,17 14 5,6 4 16,15 11 7,9 2 13,17 E-8 (base 18)


I don't rely on series as all.

Thanks for clarifying that. There would be use for a base like twelfty in branches of mathematics where fractions containing primes larger than two and three need to be represented with positional notation after a fractional point. However, large hyperdimensional symmetry groups and larger prime numbers they can contain are unlikely to be of concern to mundane calculations and being able to describe the divisions such as of angles that are likely to be encountered in the three spatial dimensions in which ordinary people believe themselves to reside. Not all symmetry groups are relevant to metrological scalability because only certain forms can fill space as tiles or bricks. I think that dozenal, with the ability to represent thirds and quarters simply, has more of what would be wanted than any other base, without undesirable qualities such as prime numbers that have more complicated constructions, or indeed no known constructions because they are too complex.

I have just a few minutes ago done a calculation to determine the angle between octahedra folded about the fourth dimension to be 120 degrees which is in radians \(2\pi/3\), a third of a turn.

I looked at the configuration matrices of the four-dimensional regular figures that are alleged to form lattices and see that there are no five-fold symmetries. The symmetries with prime factors larger than three do not show up in that way for lattices in four dimensions. One would have to go to higher dimensions than that for the prime factor five. Nevertheless, the smaller prime factors are much more abundant in the larger group \(E_8\) order number \(2^{14} 3^5 5^2 7\), which explains why this is well rounded in dozenal.