From Newsgroup: alt.beer


Wikimedia Commons - Da Vinci Vitruve (photo L. Viatour) 2,258 x 3,070 pixels, 5.81 MB:
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"Vitruvius, the architect, says in his architectural work that the measurements
of man are in nature distributed in this manner, that is 4 fingers make a palm,
4 palms make a foot, 6 palms make a cubit, 4 cubits make a man, 4 cubits make
a footstep, 24 palms make a man and these measures are in his buildings. If you
open your legs enough that your head is lowered by 1/14 of your height and raise
your arms enough that your extended fingers touch the line of the top of your
head, let you know that the center of the ends of the open limbs will be the
navel, and the space between the legs will be an equilateral triangle"

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* angle between extended middle finger tips tangent circle-square intersections
is actually 100 degrees; navel at center of circle is 1 1/2 times higher than
1/14 of man's height; angle between raised legs at calf muscle is actually 60
degrees as measured from the center of the square; also, angle between raised
legs at center of ball of foot is 60 degrees as measured from center of circle;

* the line segment between center of circle and center of square is the opposite
side of a right triangle, with adjacent side the horizontal circle radius, and
hypotenuse from the center of the square to the end of that same circle radius,
the angle of which is 80 degrees; the center of square is 2 cubits above floor
line, and its base is tangent to the base of circle at the vertical centerline;
thus solving for "y": y/(y + 2) = tan 10; y = ~0.428148 cubits; 4 cubits/14 is
~0.285714, for a ratio of ~1.49852; very nearly 1 1/2 times higher than "1/14";

* circle radius 2 + y = ~2.428148 cubits; circle diameter 2y + 4 = ~4.856296 cu-
bits; circle area (2 + y)^2 * pi = ~18.522525 square cubits; top of circle is
2y = ~0.856296 cubits above square, segment chord 4 * sqrt(2y) = ~3.701451 cu-
bits, central angle is 2 * arctan (2 * sqrt(2y)/(2 - y)) = ~99.316396 degrees
(inside edge extended middle finger tips); 1 finger is 1/24 cubit = ~0.041667
cubits; 1 palm is 1/6 cubit = ~0.166667 cubits; 1 foot 4/6 = ~0.666667 cubits;

* simplifying the value of "y", y/(y+2)=tan(10): y = 2sin(10)/(cos(10)-sin(10));
circle chord at top of square = 8sqrt(sin(10)/(cos(10)-sin(10))) = ~3.701451;
2arcTan((cos(10)-sin(10))sqrt(sin(10)/(cos(10)-sin(10)))/(cos(10)/2-sin(10)))
is central angle of top circle sector ~99.316396 degrees; top circle sector
area = pi(2sin(10)/(cos(10)-sin(10))+2)^2arcTan((cos(10)-sin(10))sqrt(sin(10)
/(cos(10)-sin(10)))/(cos(10)/2-sin(10)))/180 = ~5.109973 square cubits; top
circle segment area = pi(2sin(10)/(cos(10)-sin(10))+2)^2arcTan((cos(10)-sin
(10))sqrt(sin(10)/(cos(10)-sin(10)))/(cos(10)/2-sin(10)))/180+(8sin(10)/(cos
(10)-sin(10))-8)sqrt(sin(10)/(cos(10)-sin(10))) = ~2.200907 square cubits;

* circle chord at side of square = 2sqrt((2sin(10)/(cos(10)-sin(10))+2)^2-4) =
~2.753836; central angle of side circle sector = 2arcTan(sqrt((2sin(10)/(cos
(10)-sin(10))+2)^2-4)/2) = ~69.091629 degrees; side circle sector area = pi
(2sin(10)/(cos(10)-sin(10))+2)^2arcTan(sqrt((2sin(10)/(cos(10)-sin(10))+2)^2
-4)/2)/180 = ~3.554865 square cubits; area of side circle segment = pi(2sin
(10)/(cos(10)-sin(10))+2)^2arcTan(sqrt((2sin(10)/(cos(10)-sin(10))+2)^2-4)/
2)/180-2sqrt((2sin(10)/(cos(10)-sin(10))+2)^2-4) = ~0.801029 square cubits;

* circle chord at bottom of square = 4 cubits; central angle of bottom circle
sector = 2arcTan(2/sqrt((2sin(10)/(cos(10)-sin(10))+2)^2-4)) = ~110.908371
degrees; bottom circle sector area = pi(2sin(10)/(cos(10)-sin(10))+2)^2arc
Tan(2/sqrt((2sin(10)/(cos(10)-sin(10))+2)^2-4))/180 = ~5.706397 square cu-
bits; bottom circle segment area = pi(2sin(10)/(cos(10)-sin(10))+2)^2arcTan
(2/sqrt((2sin(10)/(cos(10)-sin(10))+2)^2-4))/180-2sqrt((2sin(10)/(cos(10)-
sin(10))+2)^2-4) = ~2.952562 square cubits;

* area of bottom square corner outside circle = -pi(2sin(10)/(cos(10)-sin(10))
+2)^2arcTan(2/sqrt((2sin(10)/(cos(10)-sin(10))+2)^2-4))/360-sqrt((2sin(10)/
(cos(10)-sin(10))+2)^2-4)+4sin(10)/(cos(10)-sin(10))+4 = ~0.626179 square
cubits; circle chord at top corner of square = sqrt((-sqrt((2sin(10)/(cos
(10)-sin(10))+2)^2-4)-2sin(10)/(cos(10)-sin(10))+2)^2+(-4sqrt(sin(10)/(cos
(10)-sin(10)))+2)^2) = ~0.245524 cubits; central angle of top square corner
circle sector = -arcTan(sqrt((2sin(10)/(cos(10)-sin(10))+2)^2-4)/2)-arcTan
((cos(10)-sin(10))sqrt(sin(10)/(cos(10)-sin(10)))/(cos(10)/2-sin(10)))+90
= ~5.795988 degrees; top square corner circle sector area = (2sin(10)/(cos
(10)-sin(10))+2)^2(-piarcTan(sqrt((2sin(10)/(cos(10)-sin(10))+2)^2-4)/2)/
360-piarcTan((cos(10)-sin(10))sqrt(sin(10)/(cos(10)-sin(10)))/(cos(10)/2-
sin(10)))/360+pi/4) = ~0.298212 square cubits; top square corner circle
segment area = -sqrt(((-sqrt((2sin(10)/(cos(10)-sin(10))+2)^2-4)-2sin(10)
/(cos(10)-sin(10))+2)^2+(-4sqrt(sin(10)/(cos(10)-sin(10)))+2)^2)(-(-sqrt
((2sin(10)/(cos(10)-sin(10))+2)^2-4)-2sin(10)/(cos(10)-sin(10))+2)^2/4-
(-4sqrt(sin(10)/(cos(10)-sin(10)))+2)^2/4+(2sin(10)/(cos(10)-sin(10))+2)
^2))/2+(2sin(10)/(cos(10)-sin(10))+2)^2(-piarcTan(sqrt((2sin(10)/(cos(10)
-sin(10))+2)^2-4)/2)/360-piarcTan((cos(10)-sin(10))sqrt(sin(10)/(cos(10)-
sin(10)))/(cos(10)/2-sin(10)))/360+pi/4) = ~0.000508 square cubits; area
of top square corner outside circle = sqrt(((-sqrt((2sin(10)/(cos(10)-sin
(10))+2)^2-4)-2sin(10)/(cos(10)-sin(10))+2)^2+(-4sqrt(sin(10)/(cos(10)-sin
(10)))+2)^2)(-(-sqrt((2sin(10)/(cos(10)-sin(10))+2)^2-4)-2sin(10)/(cos(10)
-sin(10))+2)^2/4-(-4sqrt(sin(10)/(cos(10)-sin(10)))+2)^2/4+(2sin(10)/(cos
(10)-sin(10))+2)^2))/2+(2sin(10)/(cos(10)-sin(10))+2)^2(piarcTan(sqrt((2sin
(10)/(cos(10)-sin(10))+2)^2-4)/2)/360+piarcTan((cos(10)-sin(10))sqrt(sin(10)
/(cos(10)-sin(10)))/(cos(10)/2-sin(10)))/360-pi/4)+(-2sqrt(sin(10)/(cos(10)
-sin(10)))+1)(-sqrt((2sin(10)/(cos(10)-sin(10))+2)^2-4)-2sin(10)/(cos(10)-
sin(10))+2) = ~0.014041 (= 0.0140410224358...) square cubits;

* line segment "y" is also the shortest side of a scalene triangle, with longest
side the circle radius, and adjacent side "a" 100 degrees from vertical center-
line to the end of that same circle radius; thus solving for "a": a = sqrt((-8
sin(10)cos(10)cos(70)+4(sin(10))^2)/(-2sin(10)cos(10)+1)+(2sin(10)/(cos(10)-sin
(10))+2)^2) = ~2.316912 cubits (2.31691186136...); area of triangle = sin(10)cos
(10)cos(20)/(-sin(10)cos(10)+1/2) = ~0.488455 (0.488455385956...) square cubits;

* segment "y" is shortest side of yet another, slightly smaller scalene triangle
with adjacent side "a" 110 degrees from vertical centerline, and longest side
60 degrees from the same vertical centerline; thus solving for "a": a = sqrt(3)/
(cos(10)-sin(10)) = ~2.135278 (2.13527752148...) cubits; longest side = 2sin(70)
/(cos(10)-sin(10)) = ~2.316912 (2.31691186136...) cubits, which extends 2sin(70)
/(cos(10)-sin(10))-4sqrt(3)/3 = ~0.00751078 cubits beyond intersection w/square;
area of triangle = sqrt(3)sin(10)sin(70)/(cos(10)-sin(10))^2 = ~0.429540 square
cubits (0.429540457576...); the tiny fraction of this triangle outside square is
described by shortest side = sin(10)(2/(cos(10)-sin(10))-4sqrt(3)/(3sin(70))) =
~0.00138794 cubits (0.00138793689527...); longest side = 2sin(70)/(cos(10)-sin
(10))-4sqrt(3)/3 = ~0.00751078 (0.00751078459977...) cubits; adjacent side "a":
a = sqrt(3)(1/(cos(10)-sin(10))-2sqrt(3)/(3sin(70))) = ~0.00692198 cubits (0.00
692197652921...); area of tiny triangle = (sqrt(3)sin(10)(csc(70))^2(sqrt(3)sin
(10)/3-sqrt(3)cos(10)/3+sin(70)/2)(5sqrt(3)sin(10)sin(70)/6-4sqrt(3)sin(70)cos
(10)/3+sin(10)cos(10)+2(sin(70))^2-(sin(10))^2)+(sin(10))^2(csc(70))^2(-sqrt(3)
(cos(10)-sin(10))+3sin(70)/2)(-sqrt(3)(cos(10)-sin(10))/3+sin(70)/2))/(cos(10)-
sin(10))^2 = ~0.00000451394 square cubits (0.0000045139387711...);

* segment "y" is the base of an isosceles triangle with vertex angle 160 degrees,
leg 2(sin(10))^2/(sin(20)(cos(10)-sin(10))) = ~0.217376439936 cubits, altitude
(sin(10))^2/(cos(10)(cos(10)-sin(10))) = ~0.0377470226626 cubits, and area tan
(10)(sin(10))^2/(cos(10)-sin(10))^2 = ~0.00808065625672 square cubits; segment
"y" is also the diameter of a circle, area pi(sin(10))^2/(cos(10)-sin(10))^2 =
~0.143971899424 square cubits; this small circle is centered at the midpoint of
segment "y", i.e. between the drawing's center of circle and center of square;

* a radial grid of 36 10-degree sectors centered at each endpoint of segment "y"
highlights the many triangles and quadrangles evident in this geometric study;
a layer of about 20% opacity sector color fills makes distinguishing polygons
much easier; primary colors are red, orange, yellow, green, blue, violet, and
magenta; for zodiac equivalents, the following chart includes secondary colors
and polar angles measured in degrees from earth-sun ecliptic west at 0 scorpio:

SOLID COLOR R G B C M Y K sign *
Red 255 0 0 0 100 100 0 aries 150
Red-Orange 255 64 0 0 75 100 0 taurus 180 e
Orange 255 127 0 0 50 100 0 gemini 210
Orange-Yellow 255 191 0 0 25 100 0 cancer 240
Yellow 255 255 0 0 0 100 0 leo 270 s
Green 0 255 0 100 0 100 0 virgo 300
Cyan 0 255 255 100 0 0 0 libra 330
Blue 0 0 255 100 100 0 0 scorpio 0/360 w
Blue-Violet 64 0 255 75 100 0 0 sagittarius 30
Violet 128 0 255 50 100 0 0 capricorn 60
Violet-Magenta 191 0 255 25 100 0 0 aquarius 90 n
Magenta 255 0 255 0 100 0 0 pisces 120

* the 36 decan zodiac divides each of the twelve, 30-degree zodiac sectors into
three equal parts according to the four cardinal elements: water, air, earth,
and fire; cancer, libra, capricorn, and aries, are the cardinal zodiac signs;
scorpio, aquarius, taurus, leo, are fixed; pisces, gemini, virgo, sagittarius
are mutable; thus aries decans are red, yellow, and blue-violet; taurus decans
are red-orange, green, and violet; gemini decans are orange, cyan, and violet-
magenta; cancer decans are orange-yellow, blue, and magenta; leo decans are
yellow, blue-violet, and red; virgo decans are green, violet, and red-orange;
libra decans are cyan, violet-magenta, and orange; scorpio decans are blue,
magenta, and orange-yellow; sagittarius decans are blue-violet, red, and
yellow; capricorn decans are violet, red-orange, and green; aquarius decans
are violet-magenta, orange, and cyan; completing the circle, pisces decans
are magenta, orange-yellow, and blue; this circle is the caelestial zodiac
(caelestinum firmamentum) with its fixed signs at the cardinal directions;
its equator is the earth-sun ecliptic; prime fiducial aldebaran 15 taurus;

* the terrestrial zodiac is centered in the square, with cardinal signs at the
cardinal directions of the earth's rotational axis inclined ~23.4 degrees to
the caelestial equator; as a result this oblique rotating 36-decan zodiac is
at landfall rotated 30 degrees counterclockwise to place aries 180, libra 0,
capricorn 90, and cancer 270; taurus 210, scorpio 30, aquarius 120, and leo
300; gemini 240, sagittarius 60, pisces 150, and virgo 330; in round numbers,
the terrestrial prime fiducial is the great pyramid at 0 gemini(~29tau53:35)
longitude(~6.4 miles due east to 0gem00:00) and 30 north(~29n58:45) latitude;

[end parts 1 - 6; see part 7 - 11a for continuation]
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