Essential carpentry concepts of PEYTORILL ΣΨΦ woodworking methodology include: Plumb, Level, Square, S=OA, C=AH, T=OA, and, the Pythagoran theorem: (A)squared + (B)squared = (C)squared (…in right triangles).
Carpentry’s most important angles include 90° and 45° angles and pertain to the 360° Euclidean circle as it is divided into 180° and then into 90° quadrants where Pythagorean trigonometry takes over. These conceptual forms are significantly represented by the 45°, 45°, 90°, and 3, 4, 5 triangles that students see in high school and collegiate geometry and trigonometry textbooks and as the mathematical theory of Euclid and Pythagoras proceeds into Newtonian Calculus.
And, to begin with, the euclidean circle can be divided into four 90° quadrants by the ordinate and abscissa planes (like the sagittal, lateral and coronal planes in medicine, or the longitudinal and latitudinous planes of geospatial information systems, where lateral equates to altitudinal.
Additionally, a circle, or sphere, can also be divided into five aspects of not the four 90° quadrants yet rather of five, 72° “cinquants”. This last ΣΨΦ phenomenon of PHI (Φ), or “Five” then mysteriously correlates to Pythagoras’ golden rule, 1.161, and to the pentagonal form, which may, if only here a digression, be seen as an auspicious allusion to there being a premise whereby all of the interface confusion pertaining to the competing theoretical quantum systematics found in imperial and metric methods of conversion might somehow be found to be universal.
Accordingly, and relatively, I often ponder the pentagon. Yet, for me; I have very certainly decided upon, and like to elucidate, a premise of scale pertaining to questions as to if the Imperial, or the Metric, system should be used to appropriately quantify things pertaining to ones own life, life’s requirements, and in accordance with one’s own capabilities.
Thus: I state that Imperial quantification is best for distances under approximately three 16′ Rod; even if the NFL uses a decimal Imperial yardage system that may or may not pertain to the Pythagorean Pentagon and the Golden Rule of 5, Φ.
Imperial quantification, therefore, is ideal for those measurements of less than a few Imperial Rod, and, in correlation with a scale specific to human beings that constitutes a quantification method applicability wherein Imperial mathematics is superior relative to human scale weights and measures which in carpentry terms are specifically of larger than perhaps a 64th or 32nd of an inch yet smaller than the 48′ of a few Rod as stated.
Further, Imperial fractions pertain directly to measurements of product and action where tolerances deem the margin of error to be acceptable when the erroneous result from a given process, like cutting wood, is less than approximately half of a saw blade width and where approximations of 1/2, 1/3, 1/4, 1/8, 1/16, 1/32, or 1/64 of an inch are common. And, in this regard, the Imperial system is excellent for mill floor and job site calculations where splitting distances in halves, thirds, quarters, or eighths, or, in processes of doubling, tripling or quadrupling distances and their fractional derivations, as defined in inches, feet or rods similarly lends itself to simple and intuitive multiplication and division of lengths which in pertaining calculations is typically relative to two, four, splitting and doubling.
And, similarly, note here that one imperial foot is comprised of 12″; and with Imperial mathematics built upon multiplication and division processes with, for example, scribe tools like dividers, proportional dividers and calipers, or in drafting with an architects quarters and eighths pocket reel tape, which converts measures of such as board footage to scale with a 12″ or 6″ architectural scalar ruler, and functions with Imperial yard and foot sticks, with the various squares, and with now traditional steel tape measures amongst which I find the 16′ Rod tape optimal.
So, Imperial measurements are achieved with appropriately requisite precision via a now traditional steel Rod tape measure, with yardsticks, reel tapes, square rules, speed squares, and with 12″ and 6″ combination squares. Systematically on scales of 1/4 or 1/8, these instruments can be readily tranferred to scalar sketches and drafts, and can also readily convert forestry and silvicultural field measurements from Biltmore stick cruises and Scribner Log Rule data collections into the systematics of timber harvest, forwarding, and milling, and then via drafting into the application of heavy timber post and beam, timber column and girt, engineered lumber applications in civil structures, in conventional nominal grade stud framing builds, and in the application of light duty structural steel in Civil applications. Then, from schematic drafting similar yet often more architecturally precise than for structural design; these carpentry methods, with the carpenter’s quantification instrumentation, ideation is converted into a substantive result of framing and millwork carpentry as described above.
Meanwhile, Metric quantification methods, admittedly, are better for measures often far larger, smaller, faster, or heavier than the more human few Rod approximation.
Suffice to say that the most earthen of mathematics is based upon linear and angular considerations of the 90° euclidean quadrants and that these pertain to the earth and sun.
The Ordinal and Abscissal are linear. The ordina and the abscissa are as relative to a circle or sphere as they are to the earth and the heavens. And, in cases like that of Eden, Galilean mathematics, and of a Henge; the center point, tree of life, mathematical origin, or henge set stone is encompassed by a garden wall or henge perimeter equating to the Euclidean circle, where line of sight vectors eminate from the set point in the cardinal stellar directions to and off Polaris at 90° quadrant intervals, and additionally towards solar phenomenon such as the vernal equinox, or the hibernal solstice, where set stones can be set along the line of sight vectors to dawn and dusk points where the first and last of the day’s sunlight is seen at the horizon in the matutinal and vespertine twilight hours.
Thus, in similar fashion as to how a lensatic compass is held at ones belt for dead reckoning; a surveyors transit can plot a henge and an academic telescope can align it with the heavens.
And, concerning cases of movement from that set towards, at, or upon the henge perimeter, towards, at, near or above the surface of a sphere like the earth or of an armillary representation of such, or in accordance with a trajectory relative to the philosophical mathematical origin of the theorem, to another geopositioning “henge set”, or to proximal navigational and GIS sets with discernable mass and other geospatial definitions: all mathematical quantification is relative and proximal to ones navigational set, within the perceivable perimeter of ones awareness of relative to fixes at, within or beyond tbe perimeter.
Also admittedly, there is far more to discuss here, relative to Imperial and Metric quantifications; especially as our mathematical methods evolve from Euclidean to Pythagoran philosophy and thence towards the Newtonian Scholiums of ellipses, orbits, and even of rhumb lines crossing Cartographical meridians with a star to rule the day, and a star to rule the night; being ever present.