If I We’re a Landscaper

In overhauling and expanding the code’s formal PDF, I have come to appreciate just how much space is devoted to abstract reasoning behind the applications, and how far GDCod may seem to be removed from reality.

But in reviewing the math for wave-formed landscaping recently, I found it easier to make the connection with the volume of DIRT required to realize a plan. Before showing how useful the formulas are, here’s a brief description of how they were derived:

For half waves spun about their crests or troughs to shape mounds or round berms and embankments I used the thin shell integration method on the wave equation Y = M cos X, with Y being the height, M the maximum slope, X the radius and dx the thickness of the cylinder. I treated a quarter wave section at a time and spun from both directions to account for concave rounding situations.

Landscaping Wave Volumes

After integrating the convex crest quarter wave section, I divided this generalized volume by that of the cylinder bounding it to obtain a proportionality constant. For the concave trough volume, the convex crest volume is subtracted from the same cylindrical volume and divided by same to obtain its proportionality constant.

Wave Volume Integrations

The same procedure was used for the concave crest and the convex trough, but as these are separated from the pivot, the integrating limits are shifted from π/2 to π and proportioned to the volume of the (thick) cylindrical shell bounding them. I then I found I could multiply the proportionality constants by the volume of a wave-bounding cylindrical shell in a specified situation to obtain a real waveform’s volume.

I am not absolutely certain that this is a valid approach for volumes of waves spun about axes, but it seems reasonable and the relative size of the constants derived are consistent with what one would expect. For the derived constants and formulas, check out the new PDFs when they are published in a few weeks on this website.

Wave Area

What I am more certain of is taking the area of the wave cross-section and multiplying it by the length of an architectural embankment to determine its volume. Still, it seems strange that the constant does not depend on slope but the more one studies a range of slopes the more this makes sense.

Volume is important for two simple related reasons. Obviously, when one lays out a landscaping plan, quantity of dirt required is of great importance. The second value comes at the end of the embanking procedure which commences with determining and marking the wave’s height and width and their relationship keyed to the maximum slope chosen from a small set of options.

Waveform Embanking Basics

The next task is to stake out a string to signify the maximum slope which is half way between both the height and width. The next thing is to drive pointed rods every few feet with flat pieces angled to the maximum slope secured to the rod at the height of the string and extending upward.

Waveform StakingNext the dirt is piled in such a way that the crest and trough are horizontal and the steepening curvature of slope from the crest jives with the stakes’ board slope along the string. Upon satisfying these basic conditions, any remaining dirt is placed at the most glaring deficits eyeballed. If dirt is consumed before satisfying these conditions, material shift from excess to deficits is in order.

In my mind, this procedure was the most one could do without getting into high tech instruments. Then I explained what I was doing to my father (whose livelihood entailed working with road construction engineers) and he quickly pointed out the identical mirror image nature of the top and bottom wave portions.


Straight Slope embankments with mirror waves

Properly enlightened, I saw that the dirt volume in its entirety could then be piled up in a straight slope from top to the outward boundary. Dirt could then be sluffed away from the midline and placed directly above the string with full knowledge that the two volumes are equal as should be. Much better way. To perfect the slopes, mirror guides can be cut from a sheet of plywood.

This would work well for a symmetric wave and even though such is unnecessary or even impractical in many situations, it still poses the simplest, most elegant default waveform. All that remains is to plant the ground cover decided upon and wait for the waveform to unite the abode with the earth.

Posted in Code Application, Code History, Derivations, Ground Rules | Leave a comment

Embarrassing Moments (of Inertia)

Several weeks after having started an expansion of my comprehensive GDCode PDF this past May, I was trying to verify some math formulas I had derived 2 years prior pertaining to rotational inertia, and which were posted under the title “Intrinsic Wheel Quantification”.

Spoke Moment of InertiaI remember the engaging challenge of finally putting a calculus education (received 3 decades prior) to practice. To test myself, I started with a simple spoke moment and came up with the same expression that tables stated!

However, for displaced lines perpendicular to a line extended from the axis and representative of bodal wheel triangles and hexagons, I should have caught on to how the expression arrived at was perhaps a bit more complicated than it should be, but I deemed my procedure sound and posted without verifying because no tables listed the configurations (as simple as they are), and I was not yet aware of how handy the parallel axis theorem is.

But in reviewing my work recently, I thought to use the theorem which gave me a very different expression. So I then went through the old procedure and got the same expression as before. Which one was right? The answer eluded me until studying similar problems in my Mary Boas.

Mass element on a curveIt seems the way I had specified infinitesimal mass (dm) was not up to par, and required the Pythagorean Theorem for a radius that varies with angle. Thus, I made another go at it and my answer seemed to improve in simplicity but still did not agree with what the parallel axis theorem yielded.

Perhaps I had misapplied the theorem and didn’t fully comprehend what the conditions of its use were. But after much digging, thought, and rereading, I determined that the answer it gave me was correct and something was still awry with my calculus approach. Finally, I looked at the angle I had used to set up the trigonometric relationships and discovered that although it was the right angle in a generalized sense, the reference axis I used didn’t properly account for the changing radial distance to the mass element.

Triangle Rotational Inertia

In making the correction, the complementary angle was found to be the relevant one and so ascribed, the limits of integration were changed also. Going through the problem set up accordingly, the expression arrived at agreed with what the parallel axis theorem yielded both for the triangle above and the hexagon below.

Hexagon Rotational Inertia

As I write this the thought occurred to me that the reason I made this mistake was because I had grown accustomed to using spherical coordinates in a certain way.

The reason I sweated all this was because I wanted to make another go at getting the moment of inertia for the 2/3 circular arc element hypothetically inscribed into the wheel as something of special GDCode application. Proceeding forward with confidence in how to set up the problem, the shear complexity of the trigonometric expressions stopped me  with integral tables promising no help and I was too tired to go the substitution route.

Rim Arc Rotational Inertia

So I derived the element’s center of mass. Then after deriving the moment from the center of the completed circle with constant radius, I used the parallel axis theorem in reverse to obtain the rotational inertia about the axis passing through center of mass. With that expression, I used the theorem in the conventional manner to find the moment of inertia about the wheel axis – an expression that agreed with what I had posted the first time!


Posted in Derivations, Rolling Transport | Leave a comment

geO d’code lingo

Just a few words to bring attention to the glossary of terms in the navigation bar. In developing Geocentric Design Code over the last 15 years, I have found it best to come up with special terms and phrases to make the delivery more palatable.  Although it is a work in progress (with links to illustrations in future plans), I think it poses an adequate quick reference tool for anyone seriously hoping to grasp the code – as it stands now.

Posted in Uncategorized | Leave a comment

ET Guidelines PDF Beta

This 13-page document is the last of the 7 PDF betas. As with the expansion of the previous 6 parts completed over the course of the last 2 and half years, delving deeper into Part VII took some unexpected directions that sometimes posed serious challenges to previous code assertions. But I think the larger perspective forced by such resulted in a much stronger and broadly applicable design framework. After delving into the foundations of code-consistent architectural styles, I found the heavy lifting in specifying these structures in more detail was already done.

Part VII Intro Graphic

In the second half, I hardly expected a connection between the bodal electrodynamic model and the ultimate defiance of gravity. With the final beta PDF complete, my next task is to consolidate the 7 parts into one document. The completion also signifies the end of a posting mode that served to open up my thinking on code concepts and their applications. That purpose fulfilled, upcoming post topics are now a mystery to me. But I’m guessing they will deal more with general timeless philosophy on the one hand and more contemporary events on the other.

Posted in Code Application, Code History, Extra-topographic Guidelines, Philosophic Bases | Leave a comment

Illustrated Gallery of Extra-topographic Concepts

This final gallery attached to a major code part is basically like the 6 that preceded it over the last 2 and a half years, with 12 slides intended to portray the essence of the subject matter. But instead of going about the endeavor as if it was simply a worthwhile alternative expression that happened to be easier on the brain, I found the Extra-topographic Guidelines gallery to be both as great a challenge as writing as well as being necessary to idea development. This was especially true in the attempt to model electro-dynamic behavior with bodal geometry.

Part VII Gallery Rocket

In reviewing the relevant physics and relying most heavily on the Feynman Lectures to do so, I came upon statements that not only has there never been a model for some vital phenomena, but to do so was impossible. So many hard thinking hours went into the usually relaxing effort, often leading to ambiguous or dead ends. What resulted are pictures that seem to come to the doorstep of deeper unifying explanations. My hope is that the model will spark more rigorous pursuits that interpret bode geometry more correctly. I believe there is something there.

Posted in Code Application, Extra-topographic Guidelines | Leave a comment