ARCHITECTONICS

Dr. Hossam Aboulfotouh

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* Lecturer, Architectonic Cosmic Theories and Development, Department of Architecture, Faculty of Fine Arts, Minia University, and Director, WPAHR-V, International Union of Architects-UIA. fotouh@mail.com

We observe numbers as physical systems that according to the proposed scenario of their evolution we assume they are of two distinctive types: basic and complex. To portray this scenario, we propose the following architectonic image. A basic system-Bs may be at its primary state a heterogeneously elastic spherical body-(HESB) swimming within an Euclidean plane that forms its steering ethereal structure-(SES). The size of HESB may be as tiny as one can imagine or the opposite, relative to the size of the beholder. SES is any plane in the cosmic enclosure. Due to reasons that will be discussed in section-2, SES is built by compressed ethereal fibers and any of its cross-sections is under a uniform compressive stress; thus, HESB will be under the effect of uniform distributed load. As a mutual reaction, HESB, together with SES, construct an ethereal protecting ring around HESB, making it a nucleus of a ringed-system.

As in masonry, or megalithic, arches that carry only compressive stress9^,10, within the middle third of this ring multi-paths of semi-elliptic thrusts will take place. The partial spread out of Bs 's ring will be like masonry-arches or the internal curved domain of the megalithic-arches of the ancient temples, e.g. the arches of Stonehenge, where the internal thrust of each is a parabolic curve, fig-1.

In Bs, since the applied forces, due to the ring's structure property, are directed towards the nucleus of the ringed-system, we imagine that the thrusts will be divided into two equal groups according to the direction of their motions: clockwise and anti-clockwise. These equalized and opposite motions keep the ring stationary. If SES contains other similar HESBs, under SES's compressive stress and HESBs' equal dominance, they may mutually compose a nucleus of one-ringed Bs. Since SES is not isolated from the rest of the cosmic enclosure, the ringed system may join with similar systems located in other planes (SESs) crossing its plane. Each HESB, therefore, will split into two, equally stressed, load-bearing elements: a spherical orbiter that runs on, and stirred by, a thrust and a spherical pillar that remains in the system's nucleus. We assume the equatorial perimeter of HESB equals to the total sum of the equatorial perimeters of both the orbiter and the pillar. The orbiter will act as a roller-support (foundation) for carrying the nucleus of the Bs that swims in other SES. It will join with one of the carried nucleus's pillars to form its central HESB. The load bearing status of a HESB, a pillar, an orbiter or Bs's nucleus will be discussed in section-3, as a base for determining the spin of each.

We presume the carrying ability or the "order" of such Bs denotes its name-number, which indicates either the number of its initial HESBs, its orbiters, or its pillars. If the orbiters were more than one in the ring, we presume they distribute themselves at uniform distances on the ring's median circle, being the heads of concentric-shapes, but each has its own thrust. This keeps a uniform distribution of the excess dynamic loads on the ring. The pillars may also follow the same distribution inside the Bs's nucleus. Further, we presume the orbiters of any Bs perform an equal cycle-length X (the thrust's perimeter) in the Bs's periodic time T_b. We may symbolize basic systems according to their name-numbers and carrying abilities as Bs₁, Bs₂, Bs₃, Bs₄, and Bs_n.

Spherically, the ring of Bs swimming in a compressed cosmic enclosure is the equatorial plane of an ethereal protecting orb, and matches the equator of the system's nucleus. We presume the structure and the function of latitude rings (parallel to the equatorial ring) are the same as the system's equatorial ring. Imaginary, if tilted rings exit they will have the same structure and function. Then, we presume the general rule for all rings of any orb as: since the ring's thrusts are semi-elliptic in shape, in case of the maximum permissible but compressive stress, the perihelion and the aphelion of each thrust will be at the inner and outer limits of its ring's middle-third respectively. These two limits are not identified on uniform distances as in masonry or megalithic structures9^,10 but, as will be discussed in section-3, it is based on the harmonic frequencies of their vector-radii, due to the harmonic property of SES.

We may imagine two circles generate the thrust curve, similar to those of the cycloid curve19. The generating circle we call it the thrust-driver; its center moves on the circumference of the median circle of the ring's harmonic-middle-third (HMT). The diameter's limof the thrust-driver are the limits of HMT. The thrust-driver performs two simultaneous but opposite cycles in T_b: its self-rotation around its center (e.g., clockwise) and its orbiting cycle around Bs 's nucleus (e.g., anti-clockwise), as shown in fig-1.

Then, if the orbiters of Bs₁, Bs₂, Bs₃, Bs₄ and Bs_n performed simultaneously one cycle in T_b, their total travel length will be X, 2X, 3X, 4X, and nX respectively. When spreading out, in symmetrical way, half of HMT between the perihelion and the aphelion points, half X will be its diagonal; despite the motion is in curve it jumps in linear increments; then, X is given by:

(1)

Where, r_a and r_p are the outer and inner radii of HMT respectively; thus, r_a never equals zero. For simplicity, we hereafter will put X equal 2p R, where R is the HMT's mean-radius. Logically, Bs can carry only the systems below (or in special cases similar to) its order, e.g., Bs₄ can carry four Bs₃ and their sub-systems: Bs₂ and Bs₁; but it cannot carry Bs₅.

Moreover, the joining of basic systems that have the same X size, generate a complex-system Cs in the cosmic enclosure. If Bs₄ carries four of Bs₃, then each of Bs₃ carries three of Bs₂ and each of Bs₂ carries two of Bs₁, they generate Cs₂₈. Its initial geometric form, at the joining moment, is the pyramid of four sides, fig-2. It is assembled from 41 basic systems: 1 Bs₄, 4 Bs₃, 12Bs₂, and 24 Bs₁. It contains 64 pillars and 64 orbiters. Each loaded orbiter in the system together with one pillar of the nucleus it carries make HESB like Bs₁ at its primary state. In Cs₂₈, they are 40 in total; 12 of them will be released out of the system after the joining process, living 28 HESBs as surplus for the system, fig-3. Cs₂₈ at the end will contain 24 orbiters, 24 pillars, and 28 HESBs. The system's name-number is, therefore, the number of its surplus of HESBs; it may also be symbolized as Cs₂₄₊₄ since its name-number is the total sum of its Bs₁s and the order of its base "Bs₄."

We may observe complex systems that are generated by this way as perfect numbers, which have, at their initial states, symmetrical geometric-forms in the cosmic enclosure. The perfect number of Cs₂₈ (or Cs₂₄₊₄) is generated by: (1*2*3*4)+4+12-12=28. Yet, the perfect number of Cs₆ that is generated by only Bs₁, Bs₂ and Bs₃ is: (1*2*3)+3-3=6. The idea is how to count the remaining surplus of HESBs in the system after the joining process, taking in consideration that the HESBs of Bs₂s will be released out of the system during this process, so their quantity should be subtracted. On that account, we should note here that, the other methods that were introduced by the later Pythagorean3^,4 on how to generate a perfect number lack the purpose; as they did not view numbers as physical systems. The definition of Euclid3^,5 and the later mathematicians3^,20 is only one of the cases for complex systems.

The formula for getting the perfect number of any Cs that is assembled by four different orders of basic systems, in which Bs₁ and Bs₂ are necessary inputs, is:

Cs_n = [1*2*(2+i)*(2+i+j)] + (2+i+j) (2)

Where i is any integer from 1 to 10, and j is any integer from 1 to 14, which make the maximum base for this type of complex systems being Bs₂₆. We think, Bs₂₆ is the maximum base for the complex systems that may be found in our cosmic enclosure.

Architectonically, we may observe some atoms and solar systems as perfect numbers, taking into consideration that the complete evolution of the complex systems will be discussed in detail in section-2. In atoms, the number of their systems is the number of their neutrons, i.e., the surplus of HESBs. However, from this perspective, a Hydrogen atom matches Bs₁; we think its initial state is the neutron; as beta decay generates the main components of a Hydrogen atom14^,21. Perhaps, at atomic scale, the rest of basic systems are unobserved because their lifetime is too short. Cs₂₈ (or Cs₂₄₊₄) matches the Chromium atom, where its 24 pillars are the protons and its 24 orbiters are the electrons. Cs₆ matches the Carbon atom.

Yet, many of the known atoms are imperfect numbers. During the assembling process of such Cs, e.g., an atom, if any of its Bs₁s carried another Bs₁, it increases its neutrons (HESBs). Similarly, if any of its Bs₂s carried another Bs₂, it increases both its electrons (orbiters) and protons (pillars). In both cases, the system will be imperfect; similar cases generate the array of isotopes. In some complex systems, however, the process of increasing their HESBs, due to the excess loads of Bs₁s, transforms the system's HESBs again into a perfect state. An example is Cs₁₁₇ (or Cs₁₀₄₊₁₃) that matches the Platinum atom. It is the perfect number 117=(1*2*4*13)+13+52-52. Besides, radioactive elements may be observed as complex systems still in the process of releasing the HESBs of their Bs₂s, either in assembled or disassembled forms.  

We think, for reasons will be discussed in the coming sections, that the giant size of Cs₂₈ matches our solar system. They include discussions on the complete evolution of this giant system, from its initial pyramidal form into becoming a one multi-ringed-system, as well as its intrinsic properties: its musical law and its diverse-to-whole-same tuning law. We will call it hereafter Cs_24+4.

1. Peder, Olaf, Early Physics and Astronomy, A Historical Introduction, Cambridge University press, p53 & p120 (1993).

2. Plato, Timaeus, (330 BC.). part1-paragraph-6 http://psychclassics.yorku.ca/Plato/Timaeus

3. O'connor, J J & Ropertson, E F, Perfect numbers, (2001).

Http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Perfect_numbers.html

4. McLeish, John, Numbers: From Ancient Civilizations to the Computer, Flamingo, HarperCollins, London, chapter 6, 1992.

5. Euclid, Elements, Book IX, Proposition 36, (150 AD.).

http://www.educa.fmf.uni-lj.si/java/pck/ELEMENTS/bookIX.html

6. Einstein, Albert, Relativity: The Special and General Theory, Methuen & Co. Ltd., (1924). Http://www.marxists.org/reference/archive/einstein/index.htm

7. Samuel, Eugenie, If the Universe is Spinning, Here's How to Feel it, New Scientist, p17, (July 13, 2002).

8. Mewhinney, Mike, Huge Mars-size Rocks May Have Caused Earth's Rapid Spin, News release 93-03ar, NASA Ames Research Center (Jan. 15, 1993) http://amesnews.arc.nasa.gov/pages/releasearchive.html

9. Brick Industry Association, Technical Notes on Bric, Structural Design Brick Masonry Arches. (Oct. 1967) http://www.brickinfo.org/BIN/technotes/t31a.htm

10. Leontovich, Valerian, Frames and Arches, M.S., McGraw hill, (1959).

11. Hawking, Stephen, The Universe in a Nutshell, Bantam Books, New York (2001)

12. Davies, Paul, About Time, Simon & Schuster, New York, chapters 1, 2 &4 (1995)

13. Schwarz, Patricia, the official "SuperString " wsite http://superstringtheory.com

14. Yavorsky, B & Detlaf, A,Handbook of Physics, Mir Pub, Mosc, p996, p68,& p101 (1997)

15. Rogers, Melissa J.B, Vogt, G L & Vargo, M, The Mathematics of Micro Gravity, NASA, p11-12 (1997). http://www.hq.nasa.gov/education

16. Williams, David R., Planetary fact Sheets, NASA Goddard Space Flight Center, (June 2002)

a. http://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html ,

b. http://nssdc.gsfc.nasa.gov/planetary/factsheet/index.html ,

c. http://nssdc.gsfc.nasa.gov/planetary/factsheet/galileanfact_table.html , and

d. http://nssdc.gsfc.nasa.gov/planetary/factsheet/asteroidfact.html

17. Jewitt, David, Kuiper Belt, http://www.ifa.hawaii.edu/faculty/jewitt/kb.html

18. Martin, David, Kuiper Belt, NASA, (2003) http://solarsystem.nasa.gov/planets/profile.cfm?Object=KBOs

19. Vygodsky, M., Mathematical Handbook: Higher Mathematics, Mir Publications, Moscow, p810, (1971).

20. Gioia, A. Anthony, The Theory of Numbers: An Introduction, Markham Publication Company, Chicago, p25, (1970).

21. Serway, R.A., Physics for Scientists and Engineers: With Modern Physics, fourth edition, Saunders, p1367, (2002).

22. Frequencies of Musical Notes,

a. http://www.phy.mtu.edu/~suits/notefreqs.html

b. http://www.techlib.com/references/musical_notes_frequincies.htm

23. Oulsnam S R & Brothers, B G, Mechanics of Materials and Engineers, B I Publications, Bombay, p99-106, (1962). 

Introduction

Section-2: The Law of Music of the Spheres.

Section-3: The Law of the Spinning Motion.

Section-4: Gravitation and the Music of the Spheres.

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