Entry 12a – (New Series – Diophantics/Facetronics)
The title of this entry is : “My last word on Root-Facet Gog-Magog Solids”
If you want to know what a “Gog-Magog Solid” is, read the previous entry to this “blog”, and join the “MyEngineering – Three Scale Applications Consortium” at entry-level – (i.e. as a FAN). In my pioneer disclosure to the Consortium, I define a new class of “solids” based on “root-rectangles”. They all have equal “obtuse solid-angle apeces” and therefore qualify as “NEAR-SPHERES” . The first two entries in this blog actually introduced the subject of near-spheres to an uninterested world. In particular, I gave a complete specification of the near-sphere I called “The Endodec” ( - enhanced dodecahedron ).
This “blog” as such, treats various “innovation concepts”, and all the related information is put into the Public Domain. While I would like my contributions to be recognized, acknowledged, and attributed to me, – as originator - there are millions of “blogs” on the Internet, and nearly 100 here on “Engineering.com”. Authors, like me, are free to write as they please, but alas, there is no 100%- certain way of actually attracting readers, or of making them read . . . let alone “UNDERSTAND”.
Like other innovators before me, I am a voice “crying in the Wilderness”, but what I tell you is true, and I am ready to tell you “three times”, - or as often as necessary.
All “root-rectangle Gog-Magog solids” have solid apex angles of 348 degrees – the same value as the solid-angle-apeces of Buckminster Fuller's famous “bucky-ball” - which has 60 such solid-angle-apeces. The “root-rectangle gog-magogs” also have 60 such solid-angle-apeces. Big deal – but take note : the “gog-magogs” are not all the same. There is only one specification for the “bucky-ball”: it must consist of 12 regular pentagons, edge-abutting 20 regular hexagons, and both these regular planar polygons have the same side-lengths. To constitute “root-rectangle gog-magogs”, on the other hand, there is an infinite number of possible rectangles, with dimensions ( “a” x “b” ). ALL can be used to configure gog-magogs, and these solids will have pentagons, EQUILATS ( – not hexagons !), and root-rectangles as facets. There can consequently occur an INFINITE number of distinct and different such “root-rectangle solids”. Like the “bucky-ball”, any one such solid is of itself infinitely scaleable, to contain any reasonable “target volume” while retaining the same “side-ratios”.
We are not finished with geometrical infinities : there exists at least one other infinite family of root-facet solids : directly derivable from “rectangle-as-root-facet”-solids. We identify this new and distinct family of “solids” with the following proposition :
“To every root-rectangle gog-magog, there exactly corresponds another distinctly different gog-magog solid, which has a different number of facets, apeces and edges, all such facets being differently dimensioned from those of the parent. All are derived from a TRAPEZIUM, a root-facet “shape” that differs distinctly from a “rectangle”.
This new infinite family of solids, - each member of which is directly derived from a single parent-“root-rectangle gog-magog” - can therefore be called :
THE ROOT-TRAPEZIUM GOG-MAGOG SOLIDS
To generate a single typical member of this family of geometrical solids, we can take ANY ROOT-RECTANGLE solid – either the gog or the magog of its type. For demonstration purposes, we choose the “GOG” generated by the (3 x 5)-unit rectangle. This particular “Gog”, by virtue of its 30 root-rectangles, has 120 edges – 60 of length “a” and 60 of lenth “b”, respectively butted to the “a”-side of a “void” pentagon, or the “b”-side of a “void” equilat, To begin our derivation of the new solid, we mark the mid-points of all the edges. Since there are 120 such edges, there are 120 such midpoints. Next we draw line-segments joining the midpoints of all the facets – of the 12 “a”-pentagons,and the 20 “b”-equilats, and also the mid-points belonging to both the “a” and “b”-sides of the root-rectangles. We find that each mid-point belongs both to the root-rectangle and to one of the “void” facets. By this construction, we find that we have defined a smaller pentagon inside each pentagon, a smaller equilat inside each equilat, and a “rhombic lozenge” inside each root-rectangle. We also find that these “in-facet junction-lines” are all linked to each other, systematically, to create a “meshed lattice” that not only comprises the new pentagons, equilats and lozenges, but also marks out 60 trapezia. These trapezia are positioned to prevent the other facets, the “void-polygons” - (pentagons, equilats and lozenges,) from being “side-butt” neighbours. The 60 “trapezia” are all linked together among themselves so that acute corners of the trapezia are joined to acute corners, and obtuse corners to obtuse corners, and each trapezium-side belongs also to a different “void” polygon. The “mid-point lattice” effectively consists of “root-trapezia” and voids, together forming a new “surface-mesh” solid that shares a “common volume” with the parent “root-rectangle” solid, in this instance – a “Gog”.
To complete the derivation and construction of the “root-trapezium solid”, we now have to discard the edges, facets and apeces of the original “root-rectangle gog-magog”, by expunging them from our visual imagination. If we had started with a “real model” of the parent “Gog”, our only method of distinguishing one solid from the other would have been, to colour the edge-lines of the “Gog” differently from the edge-lines of the derived “root-trapezium solid”. Even if the first “Gog” was a wire model, and all the mid-point junction line segments were new pieces of differently coloured wire, we would find it difficult to separate the two “enmeshed lattices”. Each lattice exists in its own right – but the two lattices are concurrent in space, and are more difficult to prise apart than a couple of hard-to-separate SIAMESE twins.
Fortunately, we are at liberty to effect the construction of such “root-trapezium solids” in a different way. Once we know the accurate dimensions of the derived “root-trapezium”, we can simply make 60 replicas of it, then join these replicas by appropriate angle-points – coupling together angle-points from two different trapezia for each “point-to-point” junction.
The actual side-lengths of the “root-trapezium” derived by way of such mid-point demarcations are related to the side-lengths of the parent “root-rectangle” by exact mathematical functions – part geometric, part trigonometric. For any “parent”-rectangle (“a” x “b”)-units, the respective side-lengths of the derived traapezium are these : (but see also illustration “ILLU1”)
aa = a*cos(36-degrees) bb = b/2 & s = bb/cos(arc-tan (a/b))
For our particular “DEMONSTRATION”-(3 x 5)-units root-rectangle , the derived lengths work out as :
aa = 2.427051... units bb = 2.5 units & s = 2.9154759... units
The actual derivation of these lengths is mathematically trivial, but too tedious to be repeated here. We have to note that while the “root-rectangle” dimensions were integral, the derived dimensions are irrational in two of the three cases.
Continuing our consideration of this particular demonstration solid, (which we directly generate by linking together numbers of the particular trapezium discussed,) we shall designate the “pentagons” to have side-lengths “aa”, and the equilats to have side-lengths “bb”.
This enables us to call the derived solid a “gog” of its kind, since (“aa” (pentagon sides) < “bb” (equilat sides)). The void “rhombi/equilateral lozenges”, that were generated directly from the mid-point junctions of the original “root-rectangle”, have side-lengths equal to one-half the diagonal of this rectangle – and we call this length “s”. Its actual value will be : (but see also ILLU1)
1/2(sq.rt(“a”-squared + “b”-squared)) which (exactly) equals (b/2)/cos(arc-tan(a/b))
1/2(sq.rt(9 + 25) = (2.5)/cos(30.963757...)-degrees
1/2(5.8309519...) = (2.5)/ 0.8574929...)
2.915476... = 2.915476...
We may call these equalities “exact” - but cannot write them exactly, since they are irrational numbers... however, they are exact in theory, and in practice, we shall not have need to work with all these decimals, or use any “lengths” defined to such levels of precision.
In the present case, we found that with “a” < “b”, and “aa” < “bb”, the sides of the derived small pentagons are less than the sides of the derived small equilats, which is consistent with our previous definition of what constitutes a “gog” root-facet solid : pentagon sides less han equilat sides. If, instead, we make the sides of the pentagons equal to “bb”, and the equilat-sides equal to “aa”, - by presenting the trapezium in a reversed orientation, the solid will be a “magog”. The sides of the magog's “lozenges” will still be “s”, [bb/cos(arc-tan (a/b))] and here, [2.915476...], but the “magog”- lozenges will have angles that differ from those of the gog's lozenges.
When actual models are constructed, the “lozenges” may behave in an unexpected way : in a real model, the lozenges are liable to “flex” as if hinged about one of their diagonals, thus acting like two isoscelese triangles butted together by a shared “hinge-line”. This effect becomes visible when the solids are actually modelled in three dimensions, for handling and viewing from all sides. The only manner in which a geometrical surface - made of plane facets - can be “closed about itself” to contain a volume, is by ensuring that all “edges” between the facets are angled at “hinge-lines”.
Three-dimensional geometric shapes, – e.g. solids - need to be modelled in palpable form, if they are to be properly evaluated and recognised. Photographs or perspective pictures simulating 3D, perhaps generated by a CAD-program, may fail to convince – but actual palpable models are incontrovertible.
In the course of my amateur mathematical tractations conceerning these new solids, I tried to act like a “real mathematician”, and asked myself the question :
“ Can ANY trapezium of dimensions “aa”, “bb”, and “s” - where “aa” and “bb” denote the parallel sides, and “s” denotes the two opposide “sloping” sides – serve as a “root-facet” for gog-magogs having pentagons, equilats, lozenges and trapezia, configured as described ?
I was led to the conclusion that the angle-values of the “candidate” trapezia are the sole and necessary “deciding factor” of whether a given trapezium can generate gog-magogs by the manipulations I have defined.
There will be TWO acute angles “A-A” opposite the short parallel “aa”, and TWO obtuse angles “O-B” opposite long parallel “bb”. The following equivalent conditions determine whether a given trapezium can configure a “gog-magog” - OR NOT ! - :
(60-degrees) < “A-A” < (90-degrees)
OR
equivalently, (90-degrees) < “O-B” < (120-degrees)
This means that all so-qualified trapezia can be used to generate their particular gog-magogs, without implied reference to any parent “root-rectangle”. Such qualified trapezia can be constrained by design to have simply-integral sides, or at least, sides of rational length not numbered in long strings of decimals.
In the time-honoured way of boffins disclosing mathematical truths, I leave it as an “exercise for the student” to have the above “statement-of-conditions” verified experimentally, or perhaps proved/disproved by any other means whatever.
This simplification-conclusion allowed me the option of separately defining a DEMO-construction trapezium with integer sides, instead of floundering around trying to construct the automatically-qualified trapezium discussed earlier, which was directly derived from the root-rectangle (3 x 5)-units. That derived trapezium would have sides that are difficult to measure, and angles that are impossible to reconstitute with any degree of precision. The actual trapezium I chose to construct has some of the dimensions of the “root rectangle (3 x 5)-units, all sides of which are easy integers.
The trapezia in “models” shown in the attached set of photographs & illustrations have “long” and “short” sides derived from the sides of this original (3 x 5)-unit root-rectangle. Their “sloping sides” are equal to the half-length of the (3 x 5)-diagonal which, unscaled, necessarily has the length (sq.rt 34)/2 (2.9154759...)-units. When these three lengths are multiplied by “scale-factor 12”, they yield a good “diophantic” approximation, a trapezium :
“aa” = 36, “s” = 35, & “bb” = 60
= 3 x 12 = 2.9154759 x 12 = 5 x 12
The trapezia of my photograped “model of a solid” have these scaled dimensions, and some relevant “trapezium-templates” are included in the attached set of illustrations, (as ILLU2,) to allow interested experimenters to verify all I have disclosed. They can “download” the templates, print them out on A4 paper of card-board quality, then actually duplicate the model I photographed – called ILLU3. The photograph shows a “gog”-derivative, a partial “dome” constructed with only 20 of the trapezia. A singleton root-trapezium is coupled across the openings that would produce 5 more “pentagon voids” if the complete solid was being constructed. This manipulation produces a structure which is reminiscent of the solid I photographed for a previous “blog” entry, called “ a shelter-roof”. The present model, using only 20 of the demo-construction root-trapezia, can be scaled-up further to make a good “real-life application” of the knowledge I have disclosed in this entry. Asa real structure, it could easily be adapted as a garden-pavilion, by any DIY amateur.
The same “worm of mathematical doubt” that caused me to ask the earlier question, now bit me again, to ask :
“What happens if I treat a rectangle [ a x b ] as if it was TWO trapezia, (aa = a, bb = a, s = b) and ( aa = b, bb = b, and s = a ) ?
I was pleased but not completely surprised, to verify that such a rectangle [ a x b ], manipulated as one or other of what I may call “two acting trapezia”, creates two additional “infinite families of scaleable “new-type” facetrons related to the gog-magogs. Members of the new families are closed surfaces with pentagons, equilats, angled lozenges and 60 rectangles [ a x b ], acting as one or other of the “acting root-trapezia”. However, what makes these families distinct is the fact that the pentagons and equilats have equal side-lengths. I have therefore given the name “GO-MAG” to these “new-type” facetrons, which – almost needless to say – are closed-surface solids with all apex-angles equal.
I could not leave the issue there : I “went the whole hog”, and manipulated identical “SQUARES” - firstly, as if they were acting “root-rectangles”, and secondly as if they were “acting root-trapezia”. The manipulation yielded, as the surprising result, two SINGULAR gomags – scaleable but singular, as the “bucky-ball is scaleable but singular. The first root-square-rectangle type comprised 12 pentagons, 20 equilats and 30 squares. Though a “gomag” by my definition, it is known in geometry as one of the “Archimedian surfaces”. The second type comprised 12 pentagons, 20 equilats, 30 lozenges and, - this time – 60 squares.
These various experiments and cogitations have given me several infinities of new solids to think about - and disclose here. I fondly hope that I have done some “original work” to formulate and demonstrate this piece of knowledge, but perhaps among the myriads of mathematical papers churned-out annually by professionals, “root-facet solids” have already been discovered and made known. All the facts presented in this entry were gathered by my own efforts, and I know of no popular accounts of them anywhere. The varied literature about “Buckminster-Fuller Domes” is the only body of “fact and record” devoted toa similar subject.
All these “new” solids of mine, some of which I have actually modelled and photographed, are “FACETRONS” - solids wholly defined by their plane facets. The “gog-magogs”, generated by manipulating “root-facets” : squares, rectangles, or trapezia, are brought into existence by means of a single, critical “construction method”, which I call “angle-point-coupling”. By this method, two root-facets are coupled together, one angle-point to another, by a singular mechanical means of attachment.
In practical applications, the root-facets will be real, hard, palpable artifacts : configurations of rods, tubes, girders, or sheet-materials, potentially buttressed in their interiors to give added strength. In the first stage of construction, the pentagons, equilats and lozenges will appear as VOIDS – but voids having real and palpable edges, and in effect, each such void shares edges with the root-facet it butts up against. Only in a second stage of construction, need the empty areas of the voids be filled in – and they can be filled-in with material that is less expensive, more flimsy and hence, less strong, than the material used to construct the “root-facets”. Anyone who likes to go about things the hard way, can use independently constructed versions of both the root-facets and the void-facets, then have fun building up a complete “Gog-Magod” solid, by “edge-butting” one hard item to another in a long sequence, until the job is done . . . but rather you than me.
The mechanical fittings needed to make the “point-to-point” junctions ( - equivalent to the load-bearing engine-to-waggon-to-waggon “couplings” that make up a “rail-road trains) are of critical importance : they must sustain the whole structure under applied stresses and strains. As I have previously mentioned, their design will give scope to engineer-designers seeking cost-effective ways of building human-scale gog-magog near-spheres and related structures. My “cardboard” models fudge this issue by locating a “stapling-disc” at each angle-arm concurrence, and during assembly, registring and abutting two auxiliary discs before finally clipping them together - using a stapler.
In my capacity of “innovation-engineer”, have conceived (I.e. defined and designed) several practical and inexpensive ways to effect “angle-point junctions”. Some of these could well be converted into “patentable” mechanical fittings, but I cannot afford the time or money to make patent applications or bring these designs of mine to market. I am ready assign concepts and design-rights to the highest bidder, but if none appears, posterity will have to re-invent them.
Adding illustrations to a text directly, as in a word-processed document, is not possible with this blog – although there are complex and time-consuming manipulations which allow a semblance of it. I have therefore attached the illustrations mentioned in the foregoing as “files” for the “3-Scale Applications Consortium”, freely accessible to members of the “engineering.com” website. Join the Consortium to see them, and use them as you please !
That is all I have to say on this subject – so now let applications, mine and other people's, follow as they may !