ORBITAL FLOWER ¨ IHARKA SZUCS-CSILLIK Astronomical Institute of Romanian Academy Astronomical Observatory Cluj-Napoca Str. Cires¸ilor 19, 400487 Cluj-Napoca, Romania Email:
[email protected]
Abstract. The regularizing techniques known as Kustaanheimo-Stiefel (KS) transformation have investigated. It has proved that it is very useful in n-body simulations, where it helps to handle close encounters. This paper shows how the basic transformation is a starting point for a family of polynomial coupled function. This interpretation becomes simply on writing KS transformations in quaternions form, which also helps to derive concise expressions for regularized equations of motion. Even if the KS regularization method is more easy to use, it is interesting to encapsulate the KS transformation in a family of methods, which all conserve the KS transformations’ properties. Further, an interesting point of view is considering, the orbital shapes of the restricted three-body problem (also regularized restricted three-body problem) for different initial conditions has compared with flower pattern. Key words: periodic orbits, Kustaanheimo-Stiefel, Levi-Civita, sacred geometry.
1. INTRODUCTION
Johannes Kepler believed that ”there exists a very common geometry in the universe. From universe to smallest particle of matter, everything is under violent effect of this common geometry. All branches of science follow the rules of this common geometry. This natural geometry exist every where. Where there is matter, there is geometry”. Starting from the idea that in space, nature’s letters of creation are shapes and in time, they are rhythms, we will present some interesting and beautiful periodic orbits using the generalized LC (Levi-Civita) and KS (Kustaanheimo-Stiefel) transformations. One day I got some pictures about megalithic stones from Romania to interpret some connection between sky and megalithic inscriptions. Between pictures I have seen a flower, with six symmetrical petals. Which flower is modelled, why is it so important for an ancient man to grave this symbol on a stone, or is not a flower, it is something else? Firstly, I searched a flower with 6 symmetrical petals from botanical area in Romania. I found some beautiful flowers with six petals in the Romanian fauna, but I saw interesting flowers in other place of the world and some flowers with historical connection. A lots of flowers. Secondly, I found that this flower is a sacred Romanian Astron. J. , Vol. , No. , p. 1–19, Bucharest,
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flower, the Flower of Life. In this moment, I understood much more things, but still remain one question, why the Seed of Life, an element of the Flower of Life (sacred geometry) engraved on a stone in a deep forest on a mountain?
Fig. 1 – The Seed of Life, a component of the Flower of life
In a mean time, I prepaid my presentation for the international conference ”Connection in Astronomy, Astrophysics, Space and Planetary Science” (29-30 May 2017, Cluj-Napoca), where I investigated some periodic orbits in parametric plane using LC and KS transformations. Playing with the initial conditions, I recognized that the orbits shapes have flower pattern. Could an ancient man imagined an orbit of a celestial body on the sky moving under the gravitational force and could he drew it as a flower of life? Megalithic orbital flowers? I don’t think so, but it is very interesting that using the restricted three-body problem (R3BP) and the LC and KS transformations we can obtained periodic orbits that modelled the nature or vice versa. Many ancient cultures acquired a profound knowledge of the sky independently, but the celestial bodies available for their observations were only those visible with the naked human eye (Magli, 2015; Maxim et al., 2009). The correct astronomical orientation of so many ancient megaliths leaves no doubt about the importance of astronomy to the builders. The exact function of the large stones that forms a prehistoric monument (megalith) has provoked more debate than practically any other issue in European prehistory. Over the centuries, they have variously been thought to have been used for human sacrifice, used as territorial markers, or elements of a complex ideological system, or functioned as early calendars. The developments of radiocarbon dating and dendrochronology have done much to further knowledge in this area. In Eastern Carpathians (Romania), on the Teasc Mountain in the Borsec Mountains at around 1300 m, archaeologists have uncovered several megaliths. On them we can see lines, circles, small dots, pentagrams, and modern letters made by shepherds (Lazarovici et al., 2011). It is difficult to date the symbols, but there are several similarities with Neolithic cultures’s symbols, especially with the so called Danubian writing (Merlini, 2009). At V˘atava (Mures¸ county) are stones (megaliths), one of them has engraved
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Orbital flower
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a circle, a 6-point compass rose (Fig. 1). We know that Homer sometimes used a six winds compass (Homeric system). The names going clockwise starting at north:
Fig. 2 – Stone from V˘atava
Boreas (winds from the north), Eurus (winds from the east or sunrise), Apeliotes, Notos (winds from the south), Argestes, Zephyrus (winds from the west or sunset). The five or six wind rose represented the sign of the Sun in many historical case. The six wind compass is a very old symbol in the Romanian folk art. V˘atava is at the Eastern border of the Roman Dacia province (106-275 AD), here, in mountains, lived the free Dacians. At Gura Haitii (near Vatra Dornei, Suceava county) the megalith was founded in the Paltinu river bed. On these stones are a six point wind rose too and some drawing with possible astronomical meaning.
Fig. 3 – Megalith from Gura Haitii
We mention that the Seed of Life’s pattern or six point wind rose can be seen on other megaliths too on Earth, for example on a knights templar tombstone, found within the St. Magnus cathedral in Kirkwall, Orkney.
Fig. 4 – Knights Templar tombstone from St. Magnus cathedral in Kirkwall
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2. KSn TRANSFORMATIONS
We present the generalization of KS transformation in the way of quaternion language. Remark 2.1. The fundamental formula i2 = j 2 = k 2 = ijk = −1
(1)
of the theory of quaternion came to Sir William Rowan Hamilton (1805-1865) as he was walking with his wife from Dunsink Observatory to Dublin along the Royal Canal on 16th October, 1843. He carved this formula on a stone of Broome Bridge. The known non-commutative multiplication rules of the imaginary units are obtained from equation (1) as ij = −ji = k, jk = −kj = i, ki = −ik = j.
(2)
Quaternions, introduced by Hamilton as a generalization of complex numbers, lead to a simple representation of the regularization of the spatial case of binary collisions in celestial mechanics (Waldvogel, 2006). Quaternions extend the concept of rotation in three dimensions to rotation in four dimensions. The value of the quaternion algebra is that its products are ordinary algebraic products, not the dot or cross products of standard vector algebra. The quaternion application, instead of matrix, show that the quaternion rotation operator is better, because it is singularity-free. The quaternion algebra appears to be the simplest way of studies in three dimension (Kuipers, 2002). Remark 2.2. Given the real number ul , l = 0, 1, 2, 3, the object u = u0 + iu1 + ju2 + ku3 is called a quaternion. Multiplication of quaternions is not commutative, but associative and the conjugate of a product of two quaternions is the product of the conjugates in the reverse order. Quaternions formalism applied in the regularization procedure is an easy and simplest way to describe the motion in parametric plane. Proposition 2.1. Let q = q 1 + iq 2 and Q = Q1 + iQ2 two quaternions. The n-th order LC transformation (LCn ), where n ≥ 2 we give in the following form n
n
n
q = Q = (Q1 + iQ2 )n =
