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I work with computer aided drafting software. I started my career before such software was developed. One of the courses I took to acquire my set of skills is descriptive geometry.


For those that might not be aware, descriptive geometry has it's roots in Euclidean geometry. Lines and circles drawn with the aide of straightedges and compasses. This stuff has been around for thousands of years, and is well understood. Things like how to draw two lines perpendicular to one another and how to divide a line into multiple segments provide the evidence that it works and that it is valid.


Descriptive geometry goes further than that. I do not know who discovered the means to use views to relate different perspectives of the same geometry for consideration. Following the methods taught, the relationship between two lines established in two views can be projected by a method which would demonstrate if they intersect or not. Again, the evidence of the senses, in conjunction with an understanding of the method provide validation.


These methods can be combined to render all the primitives. From simple cubes, cylinders, cones, etc, to complex manifold shapes that proliferate our world today in all the various forms of any of the objects manufactured today.


After drawing a simple cube, I can place dimensions in the appropriate views in such way that I can verify that the dimensions are equal to the geometry used to describe them. The cube is 100mm by 100mm by 100mm. Taking the appropriate scale (ruler), setting it next to the lines, look at the indexes, and it looks close enough. Is it 100mm's? It is, within say 0.25mm. It can be seen to fall on the mark easily within that precision. Can it be manufactured to exactly 100mm? Can it be manufactured within 0.25mm of 100mm? 100mm ±0.25mm. Measure the drawing, measure the part - it either is (valid) or it is not (invalid).


Model the cube on a computer. Dimension it the same way. Get the same results.


Modeling can get complex. Look at your monitor and keyboard. Chances are, somewhere in the world is a 3 dimensional model that corresponds feature for feature for each component.


Most keyboards today are comprised of multiple components. they need to be assembled. Other examples might be the wheel assembly on an automobile. In changing a tire, the lug nuts are removed from the studs, the tire is removed from the axle the spare tire is aligned over the studs and the lug nuts are tightened.


If I take three blocks, each drawn and manufactured to 100mm ±0.25mm, setting them side by side, I can measure them to see if the dimension is 300mm ±0.75mm. As long as the measurement falls between the indication marks it is good (valid). This adds up. The sum of the parts equals the whole here.


With a tire, the relationship between features can be analyzed by learning how to add and subtract the dimensions on the two interrelated components to determine if the holes will always fit over the studs. Early on, combinations of methods as the one used to assess the three blocks where developed Each of these methods rely upon and relate back to Euclidean principles developed thousands of years ago.


As designs grew more complex, it made a difference on how the dimensions where put on the print, and how the parts where measured. Communicating what features related other features became important. Placing a cube on a granite surface and checking it with a height gauge yielded three different dimensions, depending on which face was adjacent to the surface. Measuring just became much more complicated.


By differentiating the faces from one another, it could be communicated "which way is up". Labeling a feature as a datum became a means of ensuring that the intended setup could be repeated more reliably, thus ensuring that validity is maintained all the way back to the Euclidean principles it was developed from. Geometric Dimensioning and Tolerancing was introduced as a standard reference guide. Terms like flatness, roundness, parallel, perpendicular, and position were assigned symbols that could be applied to a drawing, with boxes containing letters that related back to the datums enabling more precision in communicating how to measure and evaluate the features in the models, the relationship between differing features from differing models, and correspond them to the parts in order to assess if they are valid or not.


To understand how this guide could be used, it helps to grasp how each symbol, the associated tolerance(s), and the relationship back to their respective datums relate back to Euclid. Is the circle a circle? One of the symbols, circularity, used with a clearly stated allowable tolerance identifies the scope by which a circle can vary from a circular shape and still be, within (valid) or out of (invalid) tolerance, considered a circle..


Robert E. Knapp picked a title for his book. Mathematics is About the World, So far, it's measuring up to reality as I see it. The evidence of my senses simply confirm it.

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One of the principles in Geometric Dimensioning and Tolerancing (GD&T) is referred to as Rule #1. The short version of it is "Perfect form at maximum material condition".


In conjunction with other guidelines in the specification, the cube, when mathematically considered at 100.25mm is taken as perfect. This permits calculations to be made without regard to deviations from "perfection". Drawing on discoveries in chemistry, materials can be selected, and calculations can be made to determine the maximum weight of the cube depending the material selected.


Materials come in a wide variety. Alloys, for instance, are comprised of different elements. Steel will be mandated that limit or require carbon. If up to 20% carbon is allowed, the steel can be analyzed to determine that it contains no more than 20% carbon. A range can be specified from 10% to 18% carbon. Again, metallurgists can subject the metal to different tests in order to measure the amount of carbon present. The density of the material can vary with the chemical make-up of the material. Knowing that variation permits the density to be used to calculate the weight of the cube at a given size.


In some instances, the least material condition is also used as "perfect form". Selecting the maximum density with the maximum material condition allows calculating the maximum mass. Minimum density with the minimum material condition would calculate the minimum mass. The fabricated cube can be subsequent measured by the appropriate instrument(s) to verify it is within specification.


Calculations of these types would require that both fields can be verified.


Computer modeling of parts has not provided me with confirmation at this time that complex models can compute the maximum and minimum material conditions taking into consideration the dimensioning scheme. Knowing what I know about GD&T, I have no doubts that it can and probably will be accomplished. In the meantime, the cube is modeled at 100mm, a density is selected, and the mass is determined as such.


In the latest volume of the GD&T standard, a new, somewhat controversial section has been added. The previous standard set the terms for understanding how tolerances determined the range allowed. The maximum and minimum material conditions MMC and LMC, could be referenced in different ways to provide manufacturing and inspection with "Bonus Tolerance" and "Shift". The terminology and approaches have led to differences in interpretation. The new standard has addressed this matter by providing a section to clarify the earlier ambiguity.


As a result, some controversy has arose as to which of the earlier interpretations is correct. In addition, several new tools have been added which will required also require integration, to understand how they relate to the various disciplines involved.

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On one of Robert Knapp's lectures, he addresses an issue that comes up in the office periodically. He uses a triangle for his example. Since the circle has already been used in an earlier example here, and is usually the geometric figure that comes up in the discussions,


It applies aptly here to concepts and to a lesser degree, concept formation. When is a circle not a circle?


Rand offers the notion of measurement omission as one of the elements used in concept formation. If the measurement is omitted when forming the concept circle, what is retained?

The answer to this is shape (and the process would repeat itself with shape, but I digress.)


Circularity establishes a tolerance band. Interestingly enough, GD&T applies a tolerance within a tolerance here. As a feature of size, the feature being a diameter (consider a dowel or a ring, or even a manhole cover) is given a dimensional limit that allows it to vary in size between a theoretically exact size of 1000mm and 2000mm in diameter. One possible shape that could still fulfill the dimensional requirements is an ellipse measuring 1000mm on the minor axis and 2000mm on the major, and still be in tolerance.


This might be fine for some applications. If the object produced was a tire, imagine the ride it would produce if the tires were allowed to vary by that degree in magnitude. GD&T's circularity adds a refinement to the dimensional limits. If I declare that the circularity must be maintained within 1mm, then the tire can be 1000mm on the minor with 1001 on the major, up to 1999mm on the minor and 2000mm on the major, which will be less perceptible to the passengers in the vehicle. I would submit that most folk would not notice. The tolerance of precision when drawing sheet metal is 1°. It takes a deviation greater than a degree for most folk to register, that is, perceive it.


Is the shape of the tire a circle (or for the more precision oriented among the readers, cylindrical, even though that can be debated ad nauseam.) when held to a circularity standard of 1mm? The circle conjured in the mind has a tolerance of zero. When an object is perceived that is circular in shape, and closer examination reveals that imperfections appear to exist, does that make it not circular? What is the tolerance applied? Does it matter in the context of the object being observed? Can a circularity refinement of 0 be applied. Yes. Can it be produced? I would submit a resonate "No". This is a failure to distinguish the concept from the object. Tangible objects vary, while the conceptual form takes on an invariant identity of its own. GD&T attempts to reconcile the difference by identifying along what lines - the measurement omitted must exist, but may exist within this specified range.

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What does it mean to be parallel as opposed to perpendicular?


Akin to to the philosophy of Objectivism, the notions of parallel and perpendicularity have some similarities and differences. In this section, let's consider some similarities.


To be parallel is akin to circularity. Two lines are running in the same direction are considered to be parallel, according to Euclid.

What does it mean to be parallel. Knapp does better justice in his book. I'm considering it in parallel with the GD&T philosophy drawn in contrast, or similarity to the Objectivist take on analogy.


Two line are said to be parallel, within a given limit. The cube consists of 3 parallel sets of sides. Again, setting the tolerance high, the sides can be 150mm apart, ±50mm. In measurement speak, the range can be between 100mm and 200mm. Applying the construct of parallel, using the symbol, a refinement can be specified. Parallel within 1mm. A cube can measure 100mm to 101mm between two sides, 199mm to 200mm on another two sides and 149.5mm to 150.5 on the third set of sides. The part is still valid. All three measurements fall within the specified range, within the specified refinement.


Going out on a limb here, I would submit that this is akin to an analogy. An analogy seeks to relate to a different example in kind, to aide in explaining a point in question. As an analogy, it varies from what is being explained. If it varies by too much, the point offered is in some contexts can be taken as an equivocation. It has elements that might be considered similar in one regard, but run askew in another.


The notion of perpendicular serves more as a foil. Drawing a perpendicular line to another line is clearly observable. In most examples, they cross. In others, they can be drawn short, but extending them would be shown to cross. Perpendicular defines a range of perpendicularity. To state that it exists within a given range requires more than just observation. It requires knowledge of how Euclid demonstrates how to draw a perpendicular, in conjunction with knowledge of how to draw to parallel lines within a specified distance away.


As a foil, Binswanger alludes to color in his lecture . Shades of blue (a shade being the range of being parallel to or similar to as elaborated in Abstractions from Abstractions) exist within a specified range.while yellow or red draw two different "perpendicular" contrasts to the construct. (as an elaborated range being perpendicular to or as being different from, as elaborated in Consciousness is Identification)


When discussing some of these issues with others, some find me to be esoteric in some regards, while others find the explanations quite gratifying.


(Caution . . . use at your own risk.)

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In the Inductive Way to Understand the Sky, Harriman outlines how science is taught, and why a different approach is being done at the Apple Institute. The sun is observed rising in the east, travels across the sky, and sets upon the western horizon. Standing still, on the earth, it is the sun observed in different locations in the sky as the day progresses. How do we get from this observation to the conclusion drawn that it is the earth, under our feet, that is doing the majority of the moving as a major contributing factor to adjusting the scene ever unfolding to the upward gaze.


Adding further complexity to the observations, in the summer, gazing to east, the sun rises more to the left and moves to the right as winter draws nigh in the northern hemisphere. Diverting our gaze to the west, the sun sets more to the right at the peak of summer, moving to the left until the winter solstice. This is what we see with our eyes, We do not feel the earth moving through its cycle about the sun.


At night, we gaze up at the moon. As the sun sets, we notice it is in a different position at each sunset, and sometimes not visible at all, as it starts one evening in the western sky setting shortly thereafter, and each consecutive night, it is visible closer to the eastern horizon taking longer to set, before vanishing and reappearing only to repeat the cycle.


Add to this the myriad stars that dot the sky with little points of light which look about the same, night after night, with the exception of a few that wandered, later ascertained to be planets.


To the early star gazers this always changing array, yet somehow remaining the same, was wrapped around some mysterious quality. It would not surprise me to learn that some may have even thought or wondered what this spooky action at a distance could possibly be.

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In measuring a cube, the three mutually perpendicular sides have 3 mutually parallel opposing sides. As earlier discussed, measuring the height depending on which side is set on the granite surface to be checked with the height gauge.


Observing where, on the horizon, one observes the sun to rise, depends on where one observes it from. For instance, if I  stand in my front yard, the sun may appear to rise over a pine tree, whereas if I move into the backyard, it may appear to rise over an oak tree.


In general, the direction one faces to observe the sun rise is called "east". If one stood in the same spot, day after day to observe the sunrise, they would notice that "east" consists of a range on the horizon that changed with the seasons. While this tolerance band is fine for knowing where the sun will rise, it does not work well for navigation over longer distances. Early compasses where made after someone noticed that lodestone could be used to indicate a northerly direction. The compass provided an instrument by which to measure direction providing a refinement to the tolerance band established by the sunrise.


This refinement allowed early observers to measure the angular magnitude of an event over the course of time.

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The phrase "many moons ago" referred to the passage of seasons, or years.


Early observers were able to count the number of days that passed as the moon passed from full moon to full moon, or from new moon to new moon. As the sun set on the western horizon, the number of days could be counted between the moons as well as the number of days passing for the sun to go thru the cycle established by the tolerance range observed on the western horizon. The summer and winter solstices could be identified by the sun's position on the horizon relative to the two limits observed. The passage of the moons over the from summer solstice to summer solstice or from winter solstice to winter solstice was 12 to 13 moon cycles, while the number of days was determined to be 365 to 366 days.


The passage of seasons, became known as the year, having a tolerance band of 365 to 366 days and a tolerance band of 12 to 13 moons, each referencing back to their respective phenomenon.

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In "Two, Three, Four and All That", Corvini delves into number as it pertains to counting. Starting with a shepherd determining if he had all his sheep at the end of the day that he started with at the beginning, she introduces a sack of pebbles containing one pebble for each sheep in the flock. By a process of pairing, the shepherd could ascertain this simply by moving from one pile of pebbles to another a pebble when another sheep walked by, out of the pen in the morning, and back into the pen later in the evening. In this way, he knew all the sheep were accounted for if he moved the last pebble when the last sheep walked by.


Counting is like a portable group of mental "pebbles". By identifying and arranging the numbers according to one more and one less, By assigning symbols to represent these quantities, we are able to count and retain quantities that we cannot perceive simply by looking. By establishing a group size as a standard, we are able to reduce the number of units in our counting system that we have to deal with. 1, 2, 3, 4, 5, 6, 7, 8, 9, deal with units. 10, 20, 30, etc, count groups of 10 units of the right column and regard the group of 10 as a single unit in the column to the left. The column on the left counts groups of 10 of the units in the column to its right. This greatly simplifies the process of determining which is larger. 8765 or 8762. Comparing the unit size in each column starting on the left, we simply move to the right until we see our answer.


In a greatly simplified presentation of how Corvini applies this to "the problem of applying number to magnitude", care must be taken to remember the what that we are counting. Measuring amounts to a process of counting units. How many millimeters? How many grams? How many square inches? How many cubic feet? How many minutes? How many hours? How many miles per hours? How many grams per liter? And so on.

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Longitude, by Dava Sobel


From the front cover:

Anyone alive in the eighteenth century would have known that "the longitude problem" was the thorniest scientific dilemma of the day--and had been for centuries.  Lacking the ability to measure their longitude, sailors throughout the great ages of exploration had been literally lost at sea as soon as they lost sight of land.  Thousands of lives and the increasing fortunes of nations hung on a resolution.  One man, John Harrison, in complete opposition to the scientific community, dared to imagine a mechanical solution--a clock that would keep precise time at sea, something no clock had ever been able to do on land.  Longitude is the dramatic human story of an epic scientific quest and of Harrison's forty-year obsession with building his perfect timekeeper, known today as the chronometer.  Full of heroism and chicanery, it is also a fascinating brief history of astronomy, navigation, and clockmaking, and opens a new window on our world.


Time plays an important role with regard to navigation, as well as rendering a three dimensional model. Some of the more robust CAD systems have a kinematic package that can be used to analyze motion between the components. Windshield wiper paths must be developed to keep the wiper blade on compound curved glass. Power and crank windows rely on a cylindrical shape to permit retraction into a door. Torsion bars and complex hinging mechanisms allow compact mechanisms to aid in opening and closing hoods. While time is generally not used as a factor in determining if the geometric shapes will perform as expected, it is indispensable to nautical navigation, as well as to Kepler's and Newton's investigations.


This poem references many facts involved to be taken into consideration, although not in chronological order.


Eratosthenes estimated the size of the earth using the geometric principles of his day, and developed the first map using parallels and meridians, a way of regarding a sphere intersected by parallel planes perpendicular to its axis of spin for the latitudes while another set of radially arrayed planes provided the longitudinal demarcations.1


Instruments of navigation worked together to calculate position on the waters as well as to determine directional headings. Angular measurements taken from the sun's position at noon, or by Polaris at night, determined latitude quite readily, and improvements in the instruments provided more reliable calculations to be made. Harrison's chronometer was a tale of his opposition to the scientific community of his day that thought the latitude problem could be resolved by their own.


1. I would submit the what has been described as a "triangle" drawn on curved surfaces is a misnomer.

Using orthographic projection, the perspective that permits the triangular shape on a sphere, actually is comprised of three circular arcs that only appear to be a triangle from that perspective, while the longitude and latitude demarcations are indeed circles. This has served as a flawed argument for trying to discredit Euclidean geometry, as far as I can grasp. Yes, the methods for drawing a "triangle" on curved surfaces are valid, in so far as I can tell, but a triangle still consists of three straight lines bound by the plane it is drawn upon. The other shape is derived by intersecting three planes, the intersections of which are all parallel to one, which when sited along the point view of one of the axis provide an edge view of all three planes simultaneously. This can then be projected onto any compound curvature to derive the projected intersection of the two geometries. This method ($) was developed by Edgard C. De Smet in 1934. I don't own the paper, I took the course. Perhaps he was inspired by Gauss and Riemann's work in this area, and discovered how to relate it back to Euclidean principles. This would be speculation on my part here, at this time.

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  • 6 months later...

Why the Greenwich prime meridian shifted a few hundred feet

Date: August 11, 2015

Source:University of Virginia

Summary:The prime meridian has shifted a few hundred feet. An astronomer helped figure out why.


Because the Earth is not perfectly round, and because different locations on Earth have different terrain features affecting gravitational pull, traditional ways to measure longitude have built-in variations, or errors, based on the specific location where measurements are taken. The observations were based on a vertical determined from a basin of mercury and were dependent on local conditions. However, Seidelmann said, GPS measures vertical from space in a straight line directly through the center of the Earth, effectively removing the gravitational effects of mountains and other terrain.


Traditional ways to measure longitude have built-in variations. Within the precision of the instruments used to establish the original location, it was deemed accurate to within the distance specified. The increased precision from the GPS provides for a correction that probably won't be noticeable on most wall maps of the world due to the line thickness used to illustrate them.


Another testimony to the ability of man's mind to discover, correct and/or improve his knowledge in the realm of spatial relationships.


Since the topic effectively deals with a matter of precision, earlier in the article the distance this meridian moved is referred to as either 102 meters or 334 feet. While 334 feet rounds to 102 meters, 102 meters can be rounded from 334.646 feet to 335. Of course, if the method is to simply truncate at the decimal point (pun intended).

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  • 5 years later...

Rule #1 has been around for a while. Rule #2 is being introduced as of ASME Y14.5-2018.

Perfect form at Maximum Material Condition fell under Rule #1. Exceptions to Rule #1 could be stipulated in a note (with a few exceptions as stated in the standard.)

Rule #2 gets implemented 36 years after ANSI Y14.5m-1982 had been introduced. Establishing RFS (regardless of feature size) and RMB (regardless of feature boundary) as default states explicitly a principle that had been implicit at the inception.

I graduated from high school in 1979.  I entered into the field of drafting shortly thereafter. ANSI Y14.5m-1982 was superseded by ASME Y14.5m-1994, which was superseded by ASME Y14.5m-2004, which was superseded by ASME Y14.5-2009.

I recently attended a course outlining the significant differences between ASME Y14.5-2018 and ASME Y14.5-2009.  The instructor was the same individual that had provided the significant differences between ANSI Y14.5m-1982 and ASME Y14.5-1994 roughly 25 years ago (as well as the two transitions in between.)

I look forward to seeing ASME Y14.5-2018 implemented in lieu of it's predecessor, and ASME continuing to be the front-runner to its European counterpart held under the acronym of ISO.

For ANSI/ASME and ISO, as a dimensioning philosophies, the geometry continues to serve as the basis of identity, while the standards continue to evolve toward more precise identifications.


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