#that's the magnetic field vector in spherical coordinates at the top there

[soriagravity]

physics

[biointernet]

Hourglass/Torus shape

Time, among all concepts in the world of physics, puts up the greatest resistance to being dethroned from ideal continuum to the world of the discrete, of information, of bits.... Of all obstacles to a thoroughly penetrating account of existence, none looms up more dismayingly than 'time.' Explain time? Not without explaining existence. Explain existence? Not without explaining time. To uncover the deep and hidden connection between time and existence ... is a task for the future. JOHN ARCHIBALD WHEELER, 1986

Hourglass/Torus shape

In geometry, a Torus is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution. More on Wiki. Hourglass - Torus shape Three-torus model of the universe The three-torus model is a cosmological model proposed in 1984 by Alexi Starobinski and Yakov B. Zeldovich at the Landau Institute in Moscow. The theory describes the shape of the universe (topology) as a three-dimensional torus. It is also informally known as the doughnut theory. A torus consists of a central axis with a vortex at both ends and a surrounding coherent field. Energy flows in one vortex, through the central axis, out the other vortex, and then wraps around itself to return to the first incoming vortex. The simplest description of its overall form is that of a donut, though it takes many different shapes, depending upon the medium in which it exists. For example, a smoke ring in air or a bubble ring in water are both very donut shaped. And yet an apple or an orange, which are both torus forms, are more overtly spherical.

Hourglass - Torus form Plants and trees all display the same energy flow process, yet exhibit a wide variety of shapes and sizes. Hurricanes, tornadoes, magnetic fields around planets and stars, and whole galaxies themselves are all toroidal energy systems. Extending this observation of the consistent presence of this flow form into the quantum realm, we can postulate that atomic structures and systems are also made of the same dynamic form. Torus is synonymous with electromagnetic field, Light body, Merkaba, Wei Qi field, energy bubble, or aura that makes oneness with God possible. Space and time are the framework within which the mind is constrained to construct its experience of reality. Immanuel Kant Philosophy of space and time is the branch of philosophy concerned with the issues surrounding the ontology, epistemology, and character of space and time. While such ideas have been central to philosophy from its inception, the philosophy of space and time was both an inspiration for and a central aspect of early analytic philosophy. DADA Time today

You are infinite Consciousness

A Torus energy field

Time symbolism

What is the symbol of time? 

Symbol of Time is The Hourglass

Time symbolism – What is the symbol of time? My Hourglass Collection – Time and Hourglass History and Symbolism. Welcome to MHC Virtual Museum! See also Time Philosophy and The Full History of Time https://www.myhourglasscollection.com/hourglass-torus-form/ The features of healthy living systems that Sahtouris identified are: 1. Self-creation (autopoiesis) 2. Complexity (diversity of parts) 3. Embeddedness in larger holons and dependence on them (holarchy) 4. Self-reflexivity (autognosis/self-knowledge) 5. Self-regulation/maintenance (autonomics) 6. Response-ability to internal and external stress or other change 7. Input/output exchange of matter/energy/information with other holons 8. Transformation of matter/energy/information 9. Empowerment/employment of all component parts 10. Communications among all parts 11. Coordination of parts and functions 12. Balance of Interests negotiated among parts, whole, and embedding holarchy 13. Reciprocity of parts in mutual contribution and assistance 14. Efficiency balanced by Resilience 15. Conservation of what works well 16. Creative change of what does not work well More

Human Energy Field

The human energy field forms a torus around the central energy channel. The central channel runs from the perineum to the inner most top crown of the head, just in front of the spine. The circumference of the central channel matches the circumference of the circle that is created when the tip of the thumb touches the tip of the index finger on the same hand.

Hourglass - Torus form Within the human being, each chakra, each acupuncture point, every energy center, is, in itself, a toroidal flow. It flows within itself, in both directions. Each atom, each cell, each organ, each organ system has its own toroidal field and energy flow, and each nests within the other, to create a larger, human torus. Human Energy Field (HEF) – conception about Energy Fields around Human Body. Esoteric name of HEF – Aura. The human torus connects to larger tori in the same way that the torus of a human cell or molecule connects to the larger human torus. It is part of the torus of the individual’s soul, and of the Earth, and these tori connect to the universal torus. All tori are connected to Source, which is all inclusive and all encompassing.

Hourglass/Torus shape

The shoulder is the outer waterfall, below Source and above the equator. The outer aspect is the outer most part of the field, just above and below the equator. The bowl is the lower outer aspect, between the equator and Source below. The intramatrix consists of all the layers and ‘filling’ between the central channel and the outer most aspect of the torus. The inside part of the torus. The term intermatrix applies to group fields or nested fields. It refers to the space between the individual tori of a larger toroidal system. More

Hourglass - Torus shape

Torus equation

Definition Take a hollow cylinder (a tube), bend it to a ring, connect the two open ends and you get a torus: Another construction is to revolve a circle in three dimensional space about an axis coplanar with the circle (Wikipedia). The torus is in a standard, canonical position if the circle is perpendicular to the x/y-plane and is rotated about the z-axis. Implicit Equation The implicit equation of the canonical torus with inner radius r and revolving radius R is: ( R - √ x2 + y2 ) 2 + z2 = r2 (1) This formula is derived in the figures below. Parametric Equation Torus is a 2-dimensional surface and hence can be parametrized by 2 independent variables which are obviously the 2 angles: α = angle in the x/y-plane, around the z-axis, 0° ≤ α The vector c from the origin O to the inner center C of the torus is:c(α) = (R cos α, R sin α, 0)T (2) The vector d from the inner center C of the torus to the point A on the torus surface can be written as the sum of its orthogonal components:d = cos β c1 + sin β z1 (3a) c1 = (cos α, sin α, 0)T (3b) z1 = (0, 0, 1)T (3c) The vector a = (x, y, z)T from the origin to an arbitrary point A(x, y, z) on the torus surface is:a = c + d (4) By substituting (2) and (3) into (4) we get the parametric equations of the torus: x(α, β) = (R + r cos β) cos α (5a) y(α, β) = (R + r cos β) sin α (5b) x(α, β) = z(α, β) = r sin β (5c) The parameters α, β are usually denoted by u, v, respectively. More about Torus equation: https://www.nosco.ch/mathematics/en/torus.php, http://mathworld.wolfram.com/Torus.html

Hourglass/Torus shape. Time-Space.

Hourglass/Torus shape

Three-torus model of the universe The "three-torus model of the universe", or informally "doughnut theory of the universe", is a proposed model describing the shape of the universe as a three-dimensional torus. The name comes from the shape of a doughnut, whose surface has the topology of a two-dimensional torus. Alexi Starobinski and Yakov B. Zeldovich proposed the model in 1984 from the Landau Institute in Moscow; however, the basis for his theory began much earlier than 1984. The foundation for any knowledge of the shape of the universe began in the mid-1960s with the discovery of cosmic microwave background by Bell Labs. Greater understanding of the universe's CMB provided greater understanding of the universe's topology; therefore, in a quest for cosmic understanding, NASA supported two explorer satellites, the Cosmic Background Explorer in 1989 and the Wilkinson Microwave Anisotropy Probe in 2001, which have gathered more information on CMB. Hourglass/Torus shape The Cosmic Background Explorer was an explorer satellite launched in 1989 by NASA that used a Far Infrared Absolute Spectrometer to measure the radiation of the universe. Led by researchers John C. Mather and George Smoot, COBE was able to obtain precise readings of radiation frequencies across the universe. With data on the universe’s radiation distribution, Mather and Smoot discovered small discrepancies in temperature fluctuation known as anisotropies throughout the universe. The finding of anisotropies led Mather and Smoot to conclude the universe consists of regions of varying densities. In the early stages of the universe, these denser regions of the cosmos were responsible for attracting the matter that ultimately became galaxies and solar systems. In “Microwave Background Anisotropy in a Toroidal Universe” by Daniel Stevens, Douglas Scott, and Joseph Silk of University of California Berkeley, the cosmologists proposed the isotropic universe suggests a complicated geometric structure. The researchers argued the density fluctuations reported by COBE proved “multiply connected universes are possible, the simplest is the three-dimensional torus.” Additionally, the journal concludes a torus shaped universe is compatible with COBE data if the diameter of the torus' tube is at least 80% greater than the torus’ horizontal diameter. Thus, COBE provided researchers with the first concrete evidence for a torus-shaped universe. COBE was eventually decommissioned by NASA on December 23, 1993. Hourglass/Torus shape Read the full article

[eqq-info]

Trapped Bose–Einstein condensates with quadrupole–quadrupole interactions - IOPscience

1. Introduction Inter-particle interactions in many-body systems play a key role in determining the fundamental properties of the systems. In ultracold atomic gases, neutral atoms interact through the van der Waals force, which can be described by a contact potential characterized by a single s-wave scattering length. Such a simplification results in great success in cold atomic physics.[1] For atoms possessing large magnetic moments, the long-range and anisotropic dipole–dipole interaction (DDI) may become comparable to the contact one, which leads to the dipolar quantum gases.[2] So far, the experimentally realized dipolar systems include the ultracold gases of chromium,[3] dysprosium,[4,5] and erbium[6] atoms. It is also possible to realize dipolar quantum gases with ultracold polar molecules.[7–13] Compared to the short-range and isotropic contact interactions, DDI interaction gives rise to many remarkable phenomena, such as spontaneous demagnetization,[14] d-wave collapse,[15] droplet formation,[16] and Fermi surface deformation.[17] Recently, a new quantum simulation platform based on atoms or molecules with electric quadrupole–quadrupole interaction (QQI) was theoretically proposed.[18–24] The quantum phases of quadrupolar Fermi gases in a two-dimensional (2D) optical lattice[18] and in two coupled one-dimensional (1D) pipes[20] were studied. Lahrz et al. proposed to detect quadrupolar interactions in ultracold Fermi gases via the interaction-induced mean-field shift.[19] For bosonic quadrupolar gases, Li et al. studied 2D lattice solitons with quadrupolar intersite interactions.[21] Lahrz[23] studied roton excitations of 2D quadrupolar Bose–Einstein condensates. Andreev calculated the Bogoliubov spectrum of the Bose–Einstein condensates (BECs) with both dipolar and quadrupolar interactions using non-integral Gross–Pitaevskii equation (GPE).[24] Experimentally, ultracold quadrupolar gases can potentially be realized with alkaline-earth and rare-earth atoms in the metastable 3P2 states[25–33] and homonuclear diatomic molecules.[34–37] In the present work, we explore the ground-state properties through full numerical calculations. In particular, we focus on the static properties, such as the condensate ground-state density profile and its stability. We also propose a scheme to quantitatively characterize the deformation of the condensate induced by the QQI. It is shown that, compared to the dipolar interaction, the quadrupole–quadrupole interaction can only induce a much smaller deformation due to its complicated angular dependence and short-range character. This paper is organized as follows. In Section 2, we give a brief introduction about the QQI. The formulation for the quadrupolar condensates is presented in Section 3. We then explore the ground-state properties in Section 4, with particular attention paid on the deformation and stability of the quadrupolar condensates. Finally, we conclude in Section 5. 2. Quadrupole–quadrupole interactions Here, we give a brief introduction about the QQI. As an example, we consider the classical quadrupole moment of a molecule which is described by a traceless symmetric tensor where α,β = x,y,z, (ax,ay,az) is the Cartesian coordinates of the a-th particle in the molecule, and ea is its charge. Compared to a dipole moment (a vector described by a magnitude and two polar angles), the description of a general quadrupole requires five numbers. However, the situation is greatly simplified for linear molecules or symmetric tops because there is only one independent nonzero component. Specifically, in a coordinate system with the z axis being along the molecular axis, such a molecule has Θzz = Θ, , and Θαβ = 0 (α ≠ β). After being transferred to a space-fixed coordinate system, the components of the quadrupole moment can then be expressed as where is a unit vector in the direction of the molecular axis. For two quadrupoles Θ1 and Θ2 with two molecular axes being along and , respectively, the QQI is[38] where r = |r|, , and ε0 is the vacuum permittivity. To further simplify the QQI, we assume that all particles posses the same quadrapole moment Θ and molecular axes are polarized along the z axis by an electric field gradient.[21] The QQI then reduces to where θ is the polar angle of r and is the spherical harmonic. In Fig. 1, we plot the angular dependence of Vqq(r). As can be seen, the QQI interaction is repulsive along both axial (θ = 0) and radial (θ = π/2) directions and it is most attractive along θ θm ≡ 49.1°. Compared to the dipolar interaction, the angular dependence of the quadrupolar interaction is more complicated. Moreover, the 1/r5 dependence on the inter-particle distance indicates that the QQI is a short-range interaction, while the DDI is a long-range one. It should be noted that, from the quantum mechanical point of view, because the quadrupole moment operator is of even parity, an atom with definite angular momentum quantum number J and magnetic quantum number M may carry a nonzero quadrupole moment. Consequently, one may effectively align the quadrupole moment with lights or magnetic fields to prepare the atoms in a particular angular momentum eigenstate, |J,M, or their superposition.[18,19] 3. Formulation We consider a trapped ultracold gas of N linear Bose molecules. In addition to the QQI [Eq. (4)] that was introduced in Section 2, we assume that molecules also interact via the contact interaction where a0 is the s-wave scattering length and m is the mass of the molecules. The total interaction potential then becomes The confining harmonic potential is assumed to be axially symmetric, i.e., where ω⊥ and ωz are the radial and axial trap frequencies, respectively. For convenience, we assume that the geometric average of the trap frequencies is constant. Consequently, the external trap is expressed as where λ = ωz/ω⊥ is the trap aspect ratio. Within the mean-field theory, a quadrupolar BEC is described by the condensate wave function Ψ(r,t) which satisfies the Gross–Pitaevskii equation (GPE) where , , and g0 and gq characterize the strength of the contact and quadrupolar interactions, respectively. Here, for simplicity, we have introduced the dimensionless units: for length, ωho for energy, for time, and for wave function. Consequently, , , g0 = 4πNa0/aho, gq = NΘ2/, and are all dimensionless quantities. The rescaled wave function is now normalized to unit, i.e., . From the dimensionless equation (9), it can be seen that the free parameters of the system are the trap aspect ratio λ, the contact interaction strength g0, and the quadrupolar interaction strength gq. Given that we shall only deal with the dimensionless quantities from now on, the "bar" over all variables will be dropped for convenience. The ground-state wave function can be obtained by numerically evolving Eq. (9) in imaginary time. The only numerical difficulty lies at the evaluation of the mean-field quadrupolar potential Similar to the dipolar gases, can be conveniently evaluated in the momentum space by using the convolution theorem, i.e., where and denote the Fourier and inverse Fourier transforms, respectively. Making use of the partial wave expansion , it can be easily shown that where k = |k| and . Numerically, the evaluation of can be performed using the fast Fourier transform. Finally, we fix the interaction parameters based on realistic systems. Since the s-wave scattering length is easily tunable through Feshbach resonance, here we shall only focus on the quadrupolar interaction strength gq. The quadrupole moments of the metastable alkaline-earth and rare-earth atoms,[39–43] and the ground-state homonuclear diatomic molecules[44] can be calculated theoretically. In particular, for the Yb atom and homonuclear molecules, the quadrupole moment can be as large as 30 a.u.[43,44] Therefore, for a typical configuration with N = 104, Θ = 20 a.u., ωho = (2π) 1000 Hz, and m = 150 amu, we find gq ≈ 66. As will be shown below, this QQI strength is large enough for experimental observations of the quadrupolar effects. 4. Results In this section, we investigate ground-state properties of the quadrupolar condensates. To easily identify the quadrupolar effects, we will focus on pure quadrupolar condensates by letting g0 = 0. This reduces the control parameters to λ and gq. We remark that, in the presence of the contact interaction, the results presented below remain quantitatively valid as long as gq g0. Since the QQI is partially attractive, stability is a particular important issue for the system. Numerically, it is found that, for a given λ, the condensate always becomes unstable when gq exceeds a threshold value . In Fig. 2(a), we map out the stability diagram of a quadrupolar condensate on the (λ,gq) parameter space, in which the solid line shows the critical QQI strength . The stability of a quadrupolar condensate strongly depends on the trap geometry. In fact, it becomes more stable for both highly oblate and elongated traps. This observation is in agreement with the angular distribution of the QQI shown in Fig. 1, as in both cases, the overall QQI is repulsive. To gain more insight into the stability of the quadrupolar condensates, we consider a homogeneous quadrupolar condensate, for which the dispersion relation of the collective excitations is[23] where n is the density of the gas and all quantities are in dimensionless form under the units that were previously defined. Then, by noting that the minimum value of is , equation (13) leads to an analytical expression of the critical QQI strength which suggests that a homogeneous condensate becomes unstable when . The dashed line in Fig. 2(a) represents the critical QQI strength . Here, for a given λ, the density n in Eq. (14) is taken as the numerically obtained peak condensate density corresponding to the parameters (λ,gq) on the solid line of Fig. 2(a). As can be seen, there is only a small discrepancy between the analytic and numerical stability boundaries. The inequality equation (14) underestimates the critical QQI strength because the zero-point energy in the trapping potential is neglected when we use the homogeneous result for the dispersion relation. We now turn to study how the QQI deforms the condensates. Compared to dipolar gases whose deformation is essentially characterized by a single parameter, the condensate aspect ratio,[45,46] the situation for quadrupolar condensate is more complicate. Because it is easier to visualize the deformation if the trapping potential is isotropic, we need to rescale the coordinates such that the isodensity surface of the condensate is a sphere in the absence of the QQI. For this purpose, we note that the condensate wave function at gq = 0, can be transformed into a spherically symmetric form by rescaling the coordinates according to , , and . This inspires us to consider the condensate density in the rescaled coordinates, i.e., In Figs. 2(b)–2(d), we present the contour plots of under three different pairs of parameters. Careful examination of these contour lines reveals that the condensate is stretched mainly along the radial and axial directions for λ = 1/8 and 8, respectively. While for λ = 1, the condensate is stretched roughly along the direction that is most attractive for the QQI. To characterize the deformation quantitatively, we expand at a given radius into where are the spherical coordinates for , βℓμ are the deformation parameters, and n0 is determined by . In this work, is so chosen that n0 is half of the peak condensate density. We note that because the ground-state wave function is axially symmetric, βℓμ is nonzero only if μ = 0. Figures 3(a)–3(c) show the gq dependence of the deformation parameters βℓ0 for ℓ = 2, 4, and 6 and for three different λ's. In all three cases, β60 are negligibly small. Furthermore, β20 dominates in highly anisotropic traps and β40 gives the largest contribution in isotropic potential, which is in agreement with the observation in Fig. 2. To understand these results, we express the angular dependence of the QQI, Y40(θ,), in the rescaled coordinates In Fig. 3(d), we plot for different λ's. As can be seen, for λ = 1/8 and 8 the most attractive direction is shifted to and 0.12π, respectively. Therefore, one can naturally observe that the condensates are stretched along the radial and axial directions for λ = 1/8 and 8, respectively. We note that, compared to DDI, the deformation induced by QQI is much smaller. This can be attributed to the two features of the QQI. The complicated angular distribution of the QQI makes it difficult to induce a global deformation. Moreover, the short-ranged feature makes the condensate prone to collapse. Therefore, the strong interaction regime that may be required to generate large deformation is inaccessible. In Figs. 4(4) and 4(b), we plot the gq dependence of the QQI energy, and the peak condensate density, np, respectively. As can be seen, the angular distribution of the density in an isotropic potential always makes the overall QQI attractive such that Eqq remains negative and np increases monotonically with gq. Meanwhile, in highly anisotropic potentials, the angular dependence of the density is mainly determined by the trap, which, for both elongate and oblate traps, leads to overall repulsive QQI. Consequently, the peak density decreases with growing QQI strength in the small gq region. For large gq close to the stability boundary, the gq dependence of nq depends on the value of λ, which may exhibit distinct tendency. 5. Conclusion In conclusion, we have studied the ground-state properties of a trapped quadrupolar BEC. For the geometries of the ground states, we have quantitatively characterized different components of the deformation induced by QQI. In addition, we map out the stability diagram on the (λ,gq) parameter plane. Finally, we point out that the QQI interaction strength required to induce the quadrupolar collapses is in principle accessible in, for example, a metastable Yb atom or homonuclear diatomic molecules, albeit the experimental realization of BECs of those atoms or molecules still remains challenging. Acknowledgements The computation of this work was partially supported by the HPC Cluster of ITP-CAS.