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(11) | EP 0 029 414 A1 |
| (12) | EUROPEAN PATENT APPLICATION |
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| (54) | A system to realize spherical or hemispherical or partially spherical constructions |
| (57) A system for the realization of spherical surfaces, in which a discrete number K
of spherical triangular faces of equal area, are brought close and joined to each
other along the sides, so that the sperical surface takes on the appearance of a sphere-polyhedron,
each individual triangular spherical face having only one side in common with the
adjacent face, and, in respect to the said side, the two adjacent faces are mirror
images of each other. If a sphere is completely covered with identical spherical triangular
forces, each common point formed by the apexes of a series of the spherical triangular
faces lies on a straight line formed by opposite sides of two spherical triangular
faces. Consequently the sphere-polyhedron can be obtained from a particular number
N of great circles. |
Background of the invention
[0001] The finding permits to achieve spheric or hemispheric (cupolas) surfaces or simply portions of spheric surfaces, through the device of bringing close to each other triangular spheric faces, all equal and two-by-two (chain) specular to each other, obtained, in the various cases, through the "segmentation" of the spheric surface itself by means of suitable interlacings of great circles, and it permits also to achieve the load-bearing structures of said spheric, or hemispheric (cupolas) surfaces ,or simply of portions of spheric surfaces, by the materialization of the great circles which form the various weavings or interlacings by means of continuous band-shaped constructive members (strips), which will be suitably fixed to each other.
[0002] The finding offers, furthermore, the opportunity of realizing particularly sturdy load-bearing structures for the abovesaid spheric or hemispheric (cupolas) sufaces or portions of spheric surfaces, by mea- ns of the so-called technics of the "more concentric shells" type, so that the various great circles interlacings are achieved , which interlacings are each internal to the other, being the interlacings of the same species, similar and equi-placed, and having slightly different radii , so that the interlacings themselves can be mutually anchored each other, along the sides of all the faces, by means of strenghtening latticeworks.
Summary of the invention'
I) General introduction
[0003] N continuous bandlike elements (or strips) are taken, all having the same length,and each of them is closed upon itself, in such a way to form a circle. It is essential that each of these bands, circle-shaped, be, according to the cases, suitably perforated(or anyhow marked) at well determined places, in a way which will be shortly explained, in order to obtain that this group of N (great) circles -this number N to be shortly indicated case by case - will be able to be mounted, according to a well precise scheme, such that the N great circles , joined together at the already pre-arranged perforation (or marking) places, will form, each time, a particular interlacing pattern which will constitute the load-bearing framework of a complete spheric surface, which will present the particular feature of dividing the whole sphere into a well determined number K of triangular spheric faces, all equal and specular to each other, K being a number which will be indicated in the following for the various cases.
[0004] Froma strictly constructional standpoint, each great circle represents a continuous beam closed upon itself, so that each " interlacing pattern" of the N great circles will determine, in the whole., a higly "hyperstatic"(i.e.statically indeterminable) structure.
[0005] It is, then, to be observed that, if one would desire to utilize only partially the various interlacing patterns of great circles, employing for instance, instead of some complete great circles, simply some hemi- great circles or even some portions of great circle (obtained, however, always by strips, suitably perforated or marked in that determined way), it will possible to realize, by the same abovementioned criterion (triangular spheric faces, all equal and two-by-two specular), also the load-bearing structures of hemispheric surfaces (cupolas or domes) or of portions of a spheric surface.
[0006] In the following description, only the load-bear= ing frameworks of complete spheric surfaces,will be presented, that is the "interlacing patterns" of complete great circles, since the cases of hemispheric surfaces (cupolas) or of portions of spheric surfaces are obviously included in the more general case of the complete sphere.
IT) Descriptive-schematic treatment of the four basic kinds (or species) of great circle interlacing patterns
[0007] Before dealing with the systematic treatment of the four species of basic interlacing patterns of great circles, it will be suitable to formulate a preliminar remark about the number values characterizing the sides of the spheric triangular faces which appear in each interlacing pattern.
[0008] Said sides, it will be recalled, are arcs of a great circle, and have been expressed, as it is usual, in degrees and'decimal submultiples of a degree, measured from the centre 0 of the sphere. The theoretical values have been calculated up to the eighth decimal digit, but, obviously, in the practical realizations one will stop earlier, according to what is reasonably requested by the order of magnitude of the concerned measure values.
[0009] Furthermore, always in order to emphasize the characteristics which are common to all the four basic interlacing patterns of great circles, we repeat that each interlacing pattern determines, on the sphere, spheric triangular faces all equal and such that two generic adjacent faces will have one, and only one, common side and will be specular in respect to said common side. For each interlacing ,one can think the whole sphere as obtainable by the chain-application of the specul3rity criterion which is valid for its faces, such that the sphere itself wi-11 be, in effects, conceived as a sphere-polyhedron, that is as a polyhedron having triangular spheric faces (all equal, and, if adjacent, specular to each other), which will exclusively result from the intersection of complete great circles.
[0010] This stated, we can pass to the systematic treatment of the four kinds of basic interlacing patterns of great circles.
Brief description of the drawings
[0011]
Fig. 1, which is simply of an indicative type, shows, in an axonometric (isometric) view, a materialized great circle;
Fig. 2 is a perforation plan which gives the distance between successive perforations or markings of a great circle of the type mentioned in Fig. 1, in the case of great circles which are intended for a first species of interlacing patterns;
Fig. 3 shows the- development, on a plane, of the 24 faces deriving from the first species of interlacing pattern;
Fig. 4 shows in an isometric view, a basic triangle of an interlacing pattern of the first species, with its length measuring values for each side which are expressed in degrees in respect to the sphere centre;
Figs. 5 and 6 are two perforation plans which give the distance between successive markings of a first set of three great circles,and, respectively, of a second set of six great circles for a basic interlacing pattern of the second species;
Fig. 7 shows the development on a plane of a second-species interlacing pattern;
Fig. 8 is a view similar to that of Fig. 4, but relating to a second-species interlacing pattern;
Figs. 9, 10 and 11 are views similar to those of Figs. 2, 3 and 4, but relating to a third-species interlacing pattern with fifteen great circles;
Figs. 12, 13, 14 and 15 are views similar to those of Figs. 5, 6, 7 and 8, but relating to a fourth-species interlacing pattern with n+1 great circles, Fig. 12 referring to only one of said great circules while Fig. 13 refers to the n residual ones;
Figs. 16 and 17 show generically the interlacing patterns of a cupola having the interlacing patterns according to the invention and according to the until known systems;
Figs. 18, 19 and 20 schematically illustrate the elemental structures which are obtainable according to the invention, respectively with simple, double, and triple frames (shells);
Fig. 21 shows a first derived interlacing pattern, of which
Fig. 22 represents in plan two basic second-spe= cies or third species triangles, representing the iterative portion of the derived interlacing pattern;
Figs. 23 and 24 show other schemes of derived interlacing patterns;
Fig. 25 shows, developed on a plane, the iterative portion of another derived interlacing pattern, illustrated in the
Fig. 26, completely;
Fig. 27 shows, in an isometric view, a different materialization form of a generic spheric triangle, determined by an interlacing pattern of great circles.
A) Basic first-species interlacing pattern
[0012] Let us take N = 6 constructional band-like elements (strips) and let us perforate (or anyhow mark) them as indicated in Fig. 2. Then, as usual, we will close them in a circle shape (as indicatively represented in Fig. 1) , so arranging the things that the end perforations L of each band will coincide.. Obviously, each band forming a great circle, instead of being superposed along a portion, as indicatively represented in Fig. 1, may be anyhow joined and welded, even butt welded. The quantities m and 1 appearing in Fig. 2 and characterizing the perforation (or marking) way of the band-like elements (or strips) are the following:
while it is clear that the complete sequence along the whole great circle is the following:. m̂ - 1̂ - m̂ - m̂ - 1̂ - m̂ . With the N = 6 thus perforated (or marked) grest circles, let us then realize the first-species basic in- terlacing pattern, conventionally represented in Fig. 3, from which it results that that interlacing pattern locates altogether, on the whole sphere, K = 24 faces, all equal, constituted by isosceles spheric right-angled triangles, which, by analogy with the interlacing pattern to which they belong, will be called first-species basic triangles.
[0013] Each of them, represented in Fig. 4, to which now we expressly refer, is characterized by the following data :
[0014] Obvious is the remark that the quantities 1 and m which appear in Fig. 2, with reference to the band perforations, are simply the sides of the first-spe= cies basic (or fundamental) triangle.
[0015] Returning now to Fig. 3, which represents in a conventional manner the basic first-species interlacing pattern, constituted by the N = 6 great circles, we believe that in order to achieve its exact interpretation, it will be suitable to provide some further explanation: indeed, the impossibility is known of representing on a plane, without deformations, a spherical surface, and, therefore, these graphical difficulties appear also in Fig. 3. In that Figure we have made recourse to the device of "opening" the spherical surface according to the four faces of a apherical tetrahedron, each of these faces comprising, in turn, six first-species basic triangles, as those of Fig. 4. Each face of spherical tetrahedron has been, then, in turn, "stretched" on a plane, with a certain consequent deformation. It follows from this that the K = 24 fundamental first-species spherical triangles, which form the spherical surface, and are all equal each other and all of them right-angled and isosceles, not always, however, will appear as such in the drawing, given the abovementioned graphical difficulties.
[0016] For clearness sake, then, we have indicated by the same (numbered) letter, provided once with an apex and the second time without the apex, diametrally opposed apices of the sphere-polyhedron.
[0017] Furthermore, for a reason which will appear evident in the following, when the derived interlacing patterns are considered, it is of interest to point out that the first-species interlacing pattern of Fig. 3 is characterized by the following elements:
- three axes of quaternary symmetry (pairs of diametrically opposed apices of the
sphere-polyhedron, on which 90°-angles converge), and precisely:
- four axes of ternary symmetry (pairs of diametrically opposed apices of the sphere-polyhedron,
on which 60°-angles converge), and precisely:
[0018] Incidentally, we observe that the product of the symmetry degree of an axis (the number boxed within a small rectangle) by the number of symmetry axes of that degree is a constant.
B) Basic second-species interlecing pattern
[0019] Let us take N = 9 constructional band-like elements (strips) and divide them into two groups. A first group (comprising:three bands, which we will call "first type" bands) will be perforated (or anyhow mar- ked) as indicated in Fig. 5, whereas a second group, consituted by the six residual bands (which we will call "second type" bands) will be perforated (or anyhow marked) as indicated in Fig. 6.
[0020] Then, as usual, we will close according to a circle shape all the nine bands (as indicatively represented in Fig. 1), arranging the things so that the two end holes of each band (Ca for the three first-type bands, and C for the six second-type bands) will coincide.
[0021] The quantities 1, m and c, which appear in the Figs. 5 and 6 and characterize the perforation (or marking) kind of the nine band-like elements or strips, are the following:
while, as it appears clearly from the Figs. 5 and 6 themselves, the complete sequences along the whole great circle are respectively, for the bends of the two types: m̂ - m̂ - m̂ m̂- m̂ - m̂ - m̂ - m̂ as well as 1̂- ĉ - ĉ - 1̂ - 1̂ - ĉ- ĉ - 1̂
[0022] With the N = 9 thus perforated (or marked) great circles, let us,then, realize the basic (or fundamental) second-species interlacing pattern, conventionally represented in Fig. 7, from which it appears that this interlacing pattern locates, in all, on the whole sphere, K = 48 faces, all equal, constituted by spherical right-angled triangles which, by analogy with the interlacing pattern to which they belong, will be celled "second-species fundamental triangles".
[0023] Each of them , represented in Fig. 8, to which now we refer explicitly, is characterized by the following data:
[0025] The observation appears obvious that the quantities 1,
and
which appear in the Figs. 5 and 6, with reference to the band perforations are simply the sides of the second-species fundamental triangle.
[0026] Reverting now to Fig. 7, which represents in a conventional manner the second-species basic interlacing pattern, constituted by the N = 9 great circles, we believe that, here too, in order to get its exact interpretation, it is advisable to spend some explaining words. Indeed, we are here in face of a graphical representation which is similar (but not identical) to that of Fig. 3. The only difference is that here we have had recourse to the device of "opening" the spherical surface according to the eight faces of a spherical octahedron, each of these faces comprising, in turn, six fundamental second-species triangles, as that illustrated in Fig. 8. Each octahedron face has been, then",stretched" on a plane, with a certain consequent deformation.
[0027] In order to correctly interpret the Fig. 7, it is suitable to take into account that:
- the K = 48 fundamental second-species spherical triangles are all equal and all right-angled triangles, having in mind that always exist, in the drawing, the known graphical difficulties connected to the representation of a spherical surface on a plane.
- the diametrically opposed spices of the sphere-po= lyhedron have been indicated by the same (numbered) letter, once provided with an apex and the second time without apex.
[0028] Finally, here too, for a reason which will appear evident in the following, when the derived interlacing patterns will be considered, it is of interest to point out that the second-species interlacing pattern of Fig. 7 is characterized by:
- three axes of quaternary symmetry (pairs of diametrically opposed apices of the
sphere-polyhedron,into which 45°-engles concur), and precisely:
- four axes of ternary symmetry (pairs of diametrically opposed spices of the sphere-polyhedron,
into which 66°-angles concur), and precisely:
six axes of binary simmetry (pairs of diametrically opposed apices of the sphere-polyhedron,
into which 90°-angles concur), and precisely:
[0029] Here, too, it is observed that the product of the symmetry degree of an axis (the number enclosed within the rectangular box) by the number of symmetry axes of that degree is a constant.
c) Basic third-species interlacing pattern
[0030] Let us take N = 15 band-like constructional elements (strips) and let us perforate (or anyhow mark) them as indicated in Fig. 9. Then, as usual, we will close them according to a circle shape (as indicatively represented in Fig. 1), taking care that the two end holes La of each band coincide. The quantities 1, m, and ĉ which appear in that Fig. 9 and characterize the perforation(or marking) of the fifteen band-like elements (or strips) are the following:
whereas, as it appears clearly from the same Fig. 9, the complete sequence, along the whole great circle, is the following: ĉ - 1̂ - m̂ - m̂ - 1̂ - ĉ - ĉ - 1̂ - m̂ - m̂ - 1̂ - ĉ.
[0031] With the thus perforated (or marked) N = 15 great circles, let us, then, realize the fundamental third-species interlacing pattern which is conventionally represented in Fig. 10, from which it appears that said interlacing pattern locates, in all, on the whole sphere, K = 120 faces, all equal, constituted by spherical right-angled triangles which, by analogy with the interlacing pattern to which they belong, will be called fundamental (or basic) third-species triangles.
[0032] Each of them, represented in Fig. 11, to which we now expressly make reference, is characterized by the following date:
[0033] Here, too, the observation is obvious that the quantities 1, m and c which appear in Fig. 9, in relation with the perforation of the bands are nothing but the sides of the basic third-species triangle.
[0034] Reverting then to Fig. 10, which represents in a conventional way the third-species fundamental interlacing pattern, constituted by the N = 15 great circles, we believe it will useful recall that here we have a pattern very similar to that of Fig. 7. Here, too, we have made recourse to the device of "opening" the spherical surface according to the eight faces of a spherical octahedron; the only difference is that, now, each octahedron face comprises, in turn, quite fifteen basic third-species triangles, as that of Fig. 11. Each octahedron face has been, successively, "stretched" on a plane, with a certain consequent deformation.
[0035] In order to get a correct interpretation of Fig. 10, the remarks are, therefore, valid as well as the expedients which have been already suggested with reference to Figs. 3 and 7.
[0036] In particular, when examining Fig.- 10, it is. to be taken in mind that:
- the K = 120 fundamental spherical triangles of the third-species are all equal and right-angled, even if this may not appear completely evident from the. drawing, owing to the known graphical difficulties which are encountered when representing a spherical surface on a plane;
-the diametrically opposed apices of the sphere-po= lyhedron have been indicated by an identical (numbered) letter, once provided with an apex and in the other case apice-less..
[0037] Here, too, for a reason that, as already said, will appear evident from the following, when the matter of the derived interlacing patterns will be discussed, it is of interest to point out, finally, that the third-species interlacing pattern of Fig- 10 is characterized by:
- six axes of quinary symmetry (pairs of diametrically opposed apices of the sphere-polyhedron,
into which 36°-angles concur), and precisely:
- ten axes of ternary symmetry (pairs of diametrically opposed apices of the sphere-polyhedron,
into which 60°-angles concur), and precisely:
-fifteen axes of binary symmetry (pairs of diametri-
- cally opposed apices of the sphere-po.lyhedron, into which 90°-angles concur), and
precisely:
Here, too, it is observed that the product of the degree of symmetry of an axis (the
number enclosed within the rectangular box) by the number expressing the symmetry
axes of that degree is a constant.
D) Basic fourth-species interlacing pattern
[0038] Let us consider N = n + 1 band-like constructional elements or strips (n being an integer ≥ 2) , and divide them into two groups or sets. A first set,comprising only one band (which we will call"first-type band" or "equator" will be perforated (or anyhow marked) in the way indicated in Fig. 12, while the second set, comprising the residual n bands (which we will call second-type bends) will be perforated by us (or anyhow marked) in the way indicated in Fig. 13.
[0039] Then,as usual, we will close according to a circle shape all the (n + 1) bands (as.indicatively represented in Fig. 1), taking care that the two end holes of each band (Z1 for the first-type band, and M for the n second-type bands) coincide.
[0040] The quantities 1́ and m which appear in the Figs. 12 and 13 and characterize the perforation (or marking) way of the (n + 1) band-like elements (or strips) are the following:
1̂ = 90° (that is, as we will later see,½ meridian) m̂ = 360°/2n (that is the(1/2n)
part of the equator), while, as it appears clear from the same Figs. 12 and 13, the
complete sequences a long the whole great circle, result to be, respectively for the
first-type band (equator):
and for the n second-type bands:
With the N = n + 1 thus perforated (or marked) great circles, let us, then, realize
the fourth-species fundamental interlacing pattern, which is conventionally represented
in Fig. 14.
[0041] The spherical fourth-species' interlacing pattern, formed by the (n + 1) great circles, is, substantially, the rather intuitive one which is obtained by dividing the terrestrial sphere into 2n equal spherical sectors, by means of 2n meridians (that is n great circles), all of them passing through the poles M and M'. These meridians are, in turn, intersected by the gteat circle constituting the equator, which divides them into half portians: in all, therefore, (n + 1) great circles, which will form, by intersecting each other, K = 4n faces, all equal, which are constituted by spherical doubly right-angled triangles.
[0042] We observe that the fourth-species fundamental interlacing pattern represents a rather degenerate case in respect to the three previous cases. Indeed, the D) case, rather than representing a determined resolution of the spherical surface into a well defined number K of faces, as, instead, it happens in the A), B) and C) cases, represents a whole family of resolutions of the spherical surface, as it appears clearly from the indefiniteness of the number N = (n + 1) of the great circles and of the number K = 4n of the faces. Anyway, the D) interlacing pattern, which, differently from the other three interlacing patterns, is rather intuitive, is definitely to be listed among the four fundamental interlacing patterns, both due to a completeness reason from the mathematical standpoint and to a pratical, of a constructional order, reason, since the D) interlacing pattern can find, as well as the remaining three, a full application in the case of cupolas which are realized through the technics of thenconcentric shells" with strengthening latticeworks, which we will mention in the following.
[0043] The K = 4n faces of the interlacing pattern are constituted, as it has been said, by doubly right-angled spherical triangles, all equal, which, by analogy with the interlacing pattern to which they belong, will be called "fundamental fourth-species triangles".
[0044] Each of them, represented in the Fig. 15, to which now we expressly refer, is characterized by the following data:
[0045] Here, toa, the observation is obvious that the quantities 1̂ and m̂ which appeer in the Figs. 13 and 12, with reference to the band perforations, are. nothing but the sides of the fourth-species fundamental triangle.
[0046] Reverting now to Fig. 14, which represents the fourth-species basic interlacing pattern, constituted by the N = n + 1 great circles, we observe that here we are in face of a graphical conventional representation which is similar to that which has been adopted for the other three interlacing patterns (see Figs. 3, 7 and 10). The only difference consists of the fact that here the sphere has been "opened" as a "flower" into the 2n spherical sectors obtained through the intersection of n great circles., intersecting each other along the polar axis M - M'. said spherical sectors, each containing two triangular faces, have been successively"stretched" on a plane, with a certain, consequent, deformation.
[0047] Here, too, the diametrically opposed apices of the sphere-polyhedron have been marked by (numbered) letter, once provided with an apex and, the second time, respectively apex-less.
[0048] For a correct interpretation of Fig. 14, finally, the same remarks and expedients are valid which have been suggested for the similar Figs. 3, 7 and 10, even if, in this case, the deformation errors due to stretching of the spherical triangular faces on a plane are less conspicuous.
[0049] Finally, here too, for a reason which will appear as evident in the following, when the derived interlacing patterns will be treated, it is of interest to point out that the fourth-species interlacing pattern illustrated in Fig. 14 is characterized by :
- one symmetry (M - M') axis of a 2n-degree (pair of diametrically opposed apices
of the sphere-polyhedron, into which(360°/2n)-angles concur, .so that, in conclusion,
we will have:
- n binary symmetry axes (pairs of diametrically opposed apices of the sphere-polyhedron,
into which 90°-angles concur), and precisely:
so that it follows, finally:
[0050] Here, too, it may be observed that the product of the symmetry degree of an axis (given by the number within the rectangular box) by the number of symmetry axes of that degree is a constant.
[0051] At this point, in order to conclude the systematic treatment of the four species of basic interlacing patterns of great circles, it will be suitable to formulate a general observation which joins them all together.
[0052] As it may be deduced by the Figs. 4, 8, 11 and 15, which represent the fundamental spherical triangular faces of the four species, the angles which do appear in them have only the following values:
90° ; 60° ; 45° ; 36° and 360°/2n. Now, starting from the already formulated criterion of generation of the sphere-polyhedrons, that is applying chain-wise the specularity criterion of the adjacent faces, the immediate consequence follows that the angles, pertaining to the various triangular faces, concurring into forming a given apex of the sphere-polyhedron, shall be all equal each other. And since they are always in an even number (360°' divided by 90°, or by 60°, or by 45°, or by 36° , or by 360°/2n, will give always as a result an even integer), it follows that at each apex of the sphere-po= lyhedron, if a certain angle of a certain face will be present (angle which may take only the following values: 90°, 60°, 45°, 36° or else 3-60°/2n), also its apex-opposed angle will be present. That is, an apex of the sphere-polyhedron does not constitutes a break point for the sides (arcs of a great circle) belonging to the various faces concurring to form that apex. And since this fact is valid for all the faces concurring into all the apices of the sphere-polyhedron, it follows that the resolution of the spherical surface will result, each time, from the interlacing of a well determined amount of complete great circles, intersecting each other: and a peculiarity of the finding consists quite in this fact.
III) - Further considerations of a constructional character
[0053] At this point, once concluded the systematic-des= criptive section relating to the four species of basic (or fundamental) interlacing patterns of great circles which divide the spherical surface into K spherical triangular faces, all equal each other, we will develop some considerations more closely pertaining to the constructional aspect of the structures that can thus be realized, in.the case that they are complete spherical structures as well as hemi-spheri= cal(cupolas) as well as,anyhow,portions of a spherical surface.
[0054] It is, indeed, apparent that all that has been said for the complete sphere may be quite valid also for the realization of portions of spherical surfaces, and in particular, for the hemi-spherical surfaces (cupolas).
[0055] In effects, since two great circles, intersecting each other, divide each other by half always, in order to realize a cupola, starting from the conception of a complete sphere, it will suffice to consider any base great circle, and to realize the remaining part of the structure by the half great circles all . placed on the same side in respect to the base great circle.
[0056] This stated, we believe to be suitable to insist upon one of the fundamental advantages offered by the finding, in that it is obtainable by means of a suitable interlacing pattern of great circles, that is, in practice, of continuous band-like constructional elements, all having the same length, circle-shaped and suitably perforated (or anyhow marked) for the joining with other similar bands. We will observe that the thus obtained interlacing patterns, such as they result from the the Figs. 3, 7, 10 and 14, and such as they are generally represented in the Fig. 16 (which schematically illustrates the case of a cupola), represent a novel solution which gives rise to constructions of a remarkable sturdiness, if they are compared with other spherical constructions,
[0057] which only seemingly are similar to the previous ones, as for instance could be obtained by realizing the sides of the spherical triangular faces always by (band-like) portions of a great circle, but all"independent"from each other and "broken" at the apices, where they would be simply fixed by pins or pivots with the adjacent elements, as it has been indicatively represented in Fig. 17, which schematizes a cupola which has been realized by this over-simplified criterion.
[0058] In order to convince oneself ot this, it will suffice to think about the completely different consequences which would result, in the two cases, from the complete yielding of a fastening stud or pivot at an apex. In the first case, indicatively represented in Fig. 16, owing to the continuity of the band- like strips constituting the great circles, strips which not interrupt at the apices, the consequence of the yielding of an apex pivot (for instance, in M) would be practically unimportant, and would not anyway imply a "deformability" of any part of the interlacing pattern; in the second case, instead (sides which are all independent and "interrupted" at the apices, as represented in Fig. 17), the consequence would follow now, due for instance to the yielding of the pivot at N, that all the sides concurring into this apex N, where the pivot yielded, would be now free to rotate about the pivots placed at the other end of each side (P; Q; R; S; T and U).
[0059] Briefly, we will be able to state that, in the first case (continuous great circles, according to the invention) we have a clearly hyperstatic construction; in the second case, on the contrary ("independent" sides, fixed to the apices) we have only a strictly in- deformable spherical framework.
[0060] This stated, and more and more probing the constructional aspect and the practical realization of spherical, hemi-spherical (in the case of cupolas) or partially spherical constructions, obtained according to the invention, we want to point out that it will be always possible to increase the sturdiness of the corresponding load-bearing structures through the realization of more concentric spherical surfaces, or concentric "shells",all of the same type, that is, considering the complete sphere, as that of Fig. 3, for the first-species interlacing, pattern with. K= = 24 faces, triangular and equal, or that of Fig.7, for the second-species interlacing pattern with K= =48 triangular, equal faces, or that of Fig. 10, for the third-species interlacing pattern with K = 120 triangular, equal faces, or finally that of Fig. 14, for the fourth-species interlacing pattern with K= = 4n triangular, equal faces.
[0061] Anyway, whichever be the type of interlacing pattern that one wants to achieve, here included also optionally the derived interlacing patterns, the concentric shells shall be, not only all of the same kind, but also all similar and equi-placed, in the meaning that the corresponding apices of the respective sphere-polyhedrons shall all be in line each other andin line with the (common) centre of the sphere.- Furthermore, if the various concentric shells, similar and equiplaced, will also have slightly differing each other the respective radii., it will be always feasible to join with each other, by a conventional strengthening latticework, the corresponding sides of two generic "shells", one immediately internal in respect to the other, which we will call "contiguous ahells".And this is valid for all the. "contiguous shells" of the whole, from the outermost to the innermost. In other words, the strengthening latticework will constitute a kind of seaming between the "contiguous shells", seaming which will develop itself within the gap (having a constant value) existing between the shells themselves, and along all the (equiplaced) sides of all the faces of the two said "contiguous shells".
[0062] In this way a stiff and sturdy load-bearing structure will be achieved for the whole spherical surface (or for the whole cupola, in the case of a hemi-spherical structure).
[0063] To better explain what we have just exposed about the load-bearing structures of the "more-concentric-shells" type, we have generically represented the generic triangular face (spherical) of a generic interlacing pattern of great circles, in various cases, and precisely:
- Fig.- 18: single-shell structure, that is simple- frame structure,obviously lacking in strengthening latticework;
- Fig. 19 : two-concentric-shells structure, that is a double framework structure, with a strenghtening latticework between the two contiguous (adjacent) shells;
- Fig. 20: three-concentric shells structure, that is a triple-framework structure, with two strenghtening latticeworks between the two pairs of contiguous shells which can be localized in the whole structure.
[0065] In the Figs. 19 and 20, in order to avoid useless complications in the drawing, we have represented the strengthening latticework only for a side of the generic face. Such latticework, in effects, shall be considered as extended, obviously, to all the sides of all the faces.
IV) Interlacing patterns which are derived from the fundamental interlacing patterns
[0066] Any, whichever be, of the four interlacing patterns of 1st, 2nd, 3rd or fourth species , before mentioned, which we will globally indicate by the term of fundamental (or basic) interlacing patterns, can be, in its turn, utilized as a guide or model in order to realize a new interlacing pattern, which at will can be formed by complete great circles or by parts of great circles, which will rely on the fundamental interlacing pattern, exploiting its symmetries.
[0067] It is just this novel type of interlacing pattern that we will call "derived" interlacing pattern. We believe it useful to explain, at this point, that, in practice, a derived interlacing pattern may be realized both together with the basic interlacing pattern which has generated it, and independently from it. In this second case, the basic interlacing pattern will disappear, and the derived interlacing pattern, the only one surviving the operation, may appear also much different from the basic interlacing pattern which generated it. A more careful examination will reveal, however, always that the derived interlacing pattern will present the same symmetries enjoyed by the generating fundamental interlacing pattern. Obviously, also for the derived interlacing pattern, all the considerations shall apply which were formulated for the fundamental interlacing patterns about the feasibility of the realization of concentric-shells structures.
[0068] Anyway, to further safeguard the originality and the extent of the finding, as well to further explain the concept of "derived" interlacing pattern, we want here in the following to mention at least four derived interlacing patterns, obtainable in a particularly easy way from the fundamental interlacing patterns, and chosen emblematically among the innumerable possible derived interlacing patterns.
[0069] These ere the derived interlacing patterns illustrated in Fig. 21 (first derived interlacing pattern), in the Figs. 23 and 24 (2nd end 3rd derived interlacing patterns) and in Fig. 26 (fourth derived interlacing pattern).
[0070] The first three derived interlacing patterns (Figs. 21, 23 and 24) have all the feature of being constituted by complete great circles, whereas the fourth interlacing pattern, always of the derived type,(Fig. 26),is constituted by simple arcs of great circles, and not by complete great circles.
[0071] A) The first derived interlacing pattern (Fig.21) is constituted by four complete great circles, indicated by chain lines, and it has been derived by the fundamental interlacing pattern of Fig. 7, of second-species (indicated in the background by a continuous line), by the simple joining two-by-two of the apices of the two right angles of all the pairs of second-species fundamental triangles, having the hypotenuse in common, as generically indicated in Fig. 22, wherein, for drawing clarity and expedience, it has been assumed to operate upon a plane surface instead than upon a spherical surface. From said Figure it appears anyway the
arc is perpendicular to the hypotenuse
, and since, once the computations effected, it follows also that LP = 30° (as viewed from the sphere centre), it follows that, finally, LPL= 2 x
= 2 x 30° = 60°.
[0072] Now, since the second-species triangles comprising the whole fundamental interlacing pattern are in all 48 in number, it follows that the whole development of the first derived interlacing pattern (chain lines of Fig. 21) is given by
x 48 = 30°x x 48 = 1440° = 4 x 360°, that is, as beforesaid, by four great circles.
[0073] In order to facilitate the reading of Fig. 21, we have indicated by the same number (from 1 to 4), enclosed within a small circle, the arcs (each of 60°) belonging to the same great circle.
[0074] It appecrs clear that the here examined interlacing pattern, constituted by the four great circles, marked by a chain line in Fig. 21, considered by itself alone, locates on the sphere an equilateral octshedric- hexahedric sphere-polyhedron, having in all 14 equilateral faces, six of which are square faces and eight are triangular faces, having,all of them, equal sides having each the value LPL = 60° (as viewed from the sphere centre): we have, indeed, a cube (hexahedron) and an octahedron, which are interpenetrated each other (and alternated).
[0075] At this point, it is also to be observed that all the derived interlacing patterns which will be mentioned in the following are deduced from the fundamental third-species interlacing pattern of Fig. 10, which in the drawings will be always drawn as background, by a thin continuous line (see Figs. 23 and 26). Among them, we will have:
[0076] B) The second derived interlacing pattern, indicated in Fig. 23 by a thick chain line; it is constitut- ted, in all, by six great circles, and has been deduced by the same process already adopted to deduce the first derived interlacing pattern (with the only difference that now one operates upon the fundamental third-species interlacing pattern, whereas, in the previous case, we had operated upon the fundamental second-species interlacing pattern): that is, here too, we have joined two-by-two each other the apices of the two right angles of all the pairs of basic third-species triangles having in common the hypotenuse, as it is still generally indicated in Fig. 22, wherein
appears to be always perpendicular to the hypotenuse
(assuming, always, for convenience and drawing clarity, that we operated upon a plane instead of a spherical surface). Once the computations made, this time we obtain:
(always viewed from the sphere centre).
[0077] Now, since,in all,the third-species triangles composing the whole fundamental interlacing pattern are 120 in number, it follows that the whole development of the second derived interlacing pattern (Fig. 23, thick chain line) is given by :
that is, as it has been said, six great circles.
[0078] We observe now that the here described second interlacing pattern, considered by itself alone, localizes upon the sphere an equilateral sphere-polyhedron of the icosi-dodecahedric type, having in all 32 equilateral faces, twelve of which are pentagonal end twenty are triangular, all having equal sides having each the value LPL = 36° (as viewed from the sphere centre): therefore, we have here a dodecahed- ron and an icosahedron, each other interpenetrated and alternate. Since in this case (Fig. 23) we have thought that the eye is better (than in Fig. 21) helped to reconstruct the sphere-polyhedron, we have avoided, in order not to complicate without need the drawing, to have recoprse to the device, then adopted, of marking by the same number (enclosed in a small circle) the arcs pertaining to the same great circle.
[0079] C) The third derived interlacing pattern , indicated by the dotted line, always in the same Fig. 23, is composed by ten great circles in all. It is obtainable by tracing all the possible diagonals of the pentagonal faces which appear in the previous interlacing pattern (that is, the second one). In this way twelve pentagonal Pythagoreans stars are obtained, which are distributed regularly upon the whole sphere, and which form, altogether, the third derived intere- lacing pattern.
[0080] Turning to the computations, we find that each diagonal of the pentagon, that is each side of the "star", measures 60° (always viewed from the sphere centre), so that, in all, recalling that each pentagonal star has five sides and that the pentagonal stars are in all twelwe, we will get for the whole third derived interlacing pattern a global development of: 12 x 5 x 60° = 3,600°, corresponding, as it has already been said, to ten great circles. Finally, in order to better evaluate and appreciate the two derived interlacing patterns of Fig. 23, we have thought better to draw them apart, without the basic interlacing pattern which has generated them. We have thus given in Fig. 24 the representation of a hemisphere,viewed from the top, constituted by the two said derived interlacing patterns: the second one, as usually indicated by a chain line, shows clearly (in a hemisphere) six pentagonal and ten triangular faces, while the third one, usually indicated by a dotted line, presents, always in a hemisphere , six pentagonal complete Pythagorean stars.
[0081] D) The fourth derived interlacing pattern, indicated in Fig. 26 by a more marked continuous line, differs in some way from the previously described derived patterns, since,while these were all composed of complete great circles, the present pattern is, on the contrary, composed of a set of short great circle arcs: each of them, with reference to Fig. 25, measures:
[0082] NQN = 2 x NQ = 2 x 12°,9393184 = 25°,87863688 (viewed always from the sphere centre). For convenience and clarity of drawing, it has been assumed, also in the Fig. 25, to operate upon a plane instead of a spherical surface.
[0083] The fourth derived interlacing pattern comprises altogether 62 equilateral faces (each having sidea of the value NQN=25°,87863688), of which thirty are spherical squares, twenty are spherical triangles and twelve are spherical pentagons, as it appears from the already mentioned Fig. 26.
[0084] Said interlacing pattern may be thought as generated through the following process: always with reference to Fig. 25 (comprising four basic 3rd species triangles concurring with their four 90°-angles into forming a binary-symmetry apex), let us draw, for each right angle the corresponding bisector, and consider the point N in which said bisector meets the hypotenuse Mb. Drawing finally from the point N the two
arcs, perpendicular to the two catheti of each basic triangle,we find the side of the generic face of the equilateral pattern, which, as already said, is just given by
[0085] An examination of Figs.23,24,26 may serve to nail down the already expressed concept that,in general, the derived patterns,once suppressed the basic generating interlacing pattern,may appear,at first sight, as interlacing patterns having their own individuality,i.e.conceived in an autonomous and independent way. On the basis of what has been abovesaid,this is an illusive impression,which by a more accurate check may be dispelled,through the location of the symmetry axes.
[0086] Finally, as a consideration relating to the utilization of the constructive system which is object of the invention, we want to point out that,by it,spherical, hemispherical and partially spherical bodies can be realized, apted to the most various uses, as for instance spherical containers for nuclear reactors, man-made satellites, cupolas for insulation from the external ambient for industrial plants, and so on, end this within a range of sizes which may go from the very small up to the very large one.
1. A system for the realization of spherical or hemispherical or partially spherical
constructions, in which a discrete K number of triangular spherical faces, all equal
to each other, are brought close to each other and joined, along the sides, so that
the spherical surface takes the aspect of a sphere-polyhedron, each single triangular
spherical face having in common with the adjacent face one and only one side, and,
in respect to said side, the two adjacent faces being specular to each other, finally
following,from the chain-application of said criterion of specularity, that the angles
belonging to the individual faces concurring into each of the apices of the sphere-poly=
hedron are all equal and in an even number for each apex, in such way that, if at
said apex a certain angle is present (belonging to a certain face), also its apex-opposed
angle will be present, so that an apex is not a discontinuity point for the sides
of the therein concurring faces; and, therefore, this reasoning being valid for all
the apices of the sphere-polyhedron, it follows that the sphere-polyhedron itself,
with its K spherical triangular faces, all equal and two-by-two specular, is bbtained
through the interlacing of a precise number N of complete great circles.
2. A system for the realization of a spherical construction according to Claim 1,
wherein the angles of the spherical triangular faces, which result from the fundamental
(basic) interlacing patterns of the great circles, present only the values of 90°,
60°, 45°,36° or 360°/2n, being, in particular, K = 24 for the faces having angles
α=90°, β=60° ; β=60°; K=48 for α=90°, β=60° and α=45°; K=120 for α=90°, β=60° and
α=36°; and finally K = 4n for α =90°,α=90° and β=360°/2n.
3. A system for the realization of the load-bearing structure of a spherical or hemispherical
(cupola)-or partially spherical surface, according to Claims 1 and, 2, wherein-said
load-bearing structure is constituted, each time,-by a precise interlacing pattern
of great circles (or half great-circles anchored to a base-cir= cle if the structure
in object is a cupola), and precisely by six great circles for a first type of basic
interlacing pattern, called "of 1st species", by nine great circles for a second-type of basic pattern, called "of 2nd
species", by fifteen great circles for a 3rd type of basic pattern, called "of 3rd-species"',
and by (n+1) great circles (with n being any iateger ≥ 2) for a 4th basic pattern,
called "of 4th-species", said great circles being realized through band-like constructional
elements, all of the same length, which will be closed upon themselves to form the
circle, and suitably perforated, or anyhow marked, in correspondence of the apices
of the sphere-polybedron, for the fixing with the other, therein concurring, great
circles.
4. A system for the realization of the load-bearing structure of a splierical surface,
according to Claim 3, wherein the marking scheme for the six great circles of the
1st species basic interlacing pattern is: m̂ - 1̂ - m̂ - m̂- 1̂ - m̂, m̂ being of a length
equal to 54°,73561032 and 1 equal to 70°,52877936 in respect to the sphere centre.
5. A system for the realization of the load-bearing structure of a spherical surface,according
to Claim 3, wherein the marking scheme for the three great circles of the 2nd-species
basic interlacing pattern is: m̂ - m̂ - m̂ - m̂ - m̂ - m̂ - m̂ - m̂ and for the other
six great circles is: 1̂ - ĉ - ĉ - 1̂ - 1̂ - ĉ - ĉ - 1̂ , with 1̂=54°,75561032,
ĉ=35°,26438968 and m̂=45°,00000000.
6. A system for the realization of the load-bearing structure of a spherical surface,
according to Claim 3, wherein the marking scheme for the 15 great circles of the 3rd-species
basic interlacing pattern is: ĉ - 1̂ - m̂ - m̂ - 1̂ - ĉ - ĉ -1̂ - m̂ - m̂ - 1̂
- ĉ with ĉ=20°,90515745, 1̂=37°,37736813, and m̂=31°,71747442.
7. A system for the realization of the load-bear= ing structure of a spherical surface,
according to Claim 3, wherein the marking scheme for one of the (n + 1) great circles
of the basic 4th-species interlacing pattern is:
repeated "2n times", and for the remaining n great circles is: 1̂ - 1̂ - 1̂ - 1̂ , with 1 = 90°.
repeated "2n times", and for the remaining n great circles is: 1̂ - 1̂ - 1̂ - 1̂ , with 1 = 90°.
8. A system for the realization of the load-bear= ing structure of a spherical, hemispherical
or partially spherical structure; according to Claim 3, in which the-symmetry axes
are:
- for the basic 1st-species interlacing pattern, or for those derived from it:
- three quaternary symmetry axes;
- four.ternary symmetry exes;
for the basic 2nd-species interlacing pattern, or for those deriving from it:
- three guaternary symmetry axes;
- four ternary symmetry axes;
-six binary symmetry exes;
- for the 3rd-species basic interlacing pattern, or for those derived from it:
- six quinary symmetry axes ;
- ten ternary symmetry axes ;
- fifteen binary symmetry axes ;
- for the 4th-species basic interlacing pattern, or for those derived from it:
-one 2n -degree symmetry axe;
- n binary symmetry axes .
9. A system for the realization of particularly. sturdy load-bearing structures of
spherical or hemi- spherical (cupolas) or partially spherical surfaces, according
to Claims 1 to 8, wherein more concentric "shells", similar and equi-placed are arranged,
hav-_ ing the corresponding apices of the respective sphere-polyhedrons all aligned
with each other and with the common centre of the spheres, and these shells are joined
each to other along all the sides of all the faces of the interlacing pattern, with
strengthening latticeworks, and other anchorage means.
10.A system for the realization of particularly sturdy load-bearing structures of
spherical, or hemi- spherical (cupolas), or partially spherical surfaces, wherein
more concentric "shells", similar and equi- placed,are arranged, having the corresponding
apices of the respective sphere-polyhedrons all aligned with each other and with the
common centre of the spheres, and these shells are joined each to other along all
the great circles of the interlacing pattern, with strengthening latticeworks, and
other anchoring means.
11. A. system for the realization of spherical or hemispherical (cupolas) interlacing
patterns of great circles (or parts of great circles) according to Claims 1 to 9,
wherein the band-like elements are constituted by metal girders having a curvilinear
axis and completely "reticular".
12. Interlacing patterns of spheric great circles, or hemispheric (cupolas) or partially
spheric great circles, according to Claims 1 to 8, realized in reinforced concrete
members.
13.A system for the realization of spheric or hemispheric or partially spheric interlacing
patterns, according to Claims 1 to 8, wherein the edges of the respective spherical
faces are joined together, said edges or rims beings realized with plane arcs of circular
sectors, concentric with the sphere, suitably bent or welded at the apices, in such
way to shape the face of the sphere-polyhedron.
14. A system for the realization of polyhedrons having plane (flat) faces , derived
from the interlacing patterns mentioned in the Claims 1 to 8, wherein the arcs of
great circle which constitute the sides of the individual faces of the sphere-polyhedrons
are substituted by the chords subtending said arcs.