“THEOREM: if G is a locally euclidean, connected, simply connected topological group of dimension n greater than one, then G contains a closed proper subgroup of positive dimension.”

[A theorem on locally euclidean groups, Annals of Mathematics, 1947, 650–658, 10.2307/1969132]

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Deane Montgomery2

American mathematician 1909–1992

Constantinos Apostolou Doxiadis (1914–1975) Greek architect

Source: Building Entopia - 1975, Chapter 2, The great danger, p. 21

“Supergravity theories generically contain non-compact global symmetry groups. The general rule is that the scalar fields of the theory in question parametrize a symmetric space. Thus, if the non-compact symmetry group is G, and its maximal compact subgroup is H, the scalar fields map the space-time into the symmetric space G/H, and the number of scalar fields is dim G – dim H. The first supergravity example of this type to be found, N = 4 supergravity is one of the most interesting. In this case there are two scalar fields and the symmetric space is SL(2,R)/SO(2).”

John Henry Schwarz (1941) American theoretical physicist

p. 1

“A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension.”

Benoît Mandelbrot book The Fractal Geometry of Nature

The Fractal Geometry of Nature (1982)

“(Jokingly) Sex in ten dimensions is impossible… topologically.”

Leonard Susskind (1940) American physicist

Lecture "Cosmic landscape and illusion of intelligent design", DESY Hamburg (28 September, 2006).

“A solid stores energy in the vibrations of the atoms about their equilibrium positions. In the simplest approximation, which Einstein considered, each atom vibrates with the same frequency, ν, in each of three dimensions, so a solid containing N atoms can be thought of as 3N one-dimensional simple harmonic oscillators.”

Jim Peebles (1935) Canadian-American astronomer

[Quantum Mechanics, 1992, Princeton University Press, 11, https://books.google.com/books?id=hTaw4Ht4Vi8C&pg=PA11]

“Calculation establishes that a quantum positive energy theorem may be possible using a representation based on the Ashtekar connection. Left open is a key question of whether this use of the Ashtekar connection is necessary or whether a positive quantum energy result can be achieved for representations based on other connections, i. e., for [other] values of the Immirzi parameter.”

Lee Smolin (1955) American cosmologist

"Positive energy in quantum gravity" arXiv (Jun 10, 2014)

“Euclidean geometry is only one of several congruence geometries… Each of these geometries is characterized by a real number K, which for Euclidean geometry is 0, for the hyperbolic negative, and for the spherical and elliptic geometries, positive. In the case of 2-dimensional congruence spaces… K may be interpreted as the curvature of the surface into the third dimension—whence it derives its name…”

Howard P. Robertson (1903–1961) American mathematician and physicist

Geometry as a Branch of Physics (1949)

“Love, Allie concluded, wasn't blind, it simply saw alternate dimensions.”

Neal Shusterman book Everfound

Source: Everfound

“It is well known that in three-dimensional elliptic or spherical geometry the so-called Clifford's parallelism or parataxy has many interesting properties. A group-theoretical reason for the most important of these properties is the fact that the universal covering group of the proper orthogonal group in four variables is the direct product of the universal covering groups of two proper orthogonal groups in three variables. This last-mentioned property has no analogue for orthogonal groups in n (>4) variables. On the other hand, a knowledge of three-dimensional elliptic or spherical geometry is useful for the study of orientable Riemannian manifols of four dimensions, because their tangent spaces possess a geometry of this kind.”

Shiing-Shen Chern (1911–2004) mathematician (1911–2004), born in China and later acquiring U.S. citizenship; made fundamental contributio…

[On Riemannian manifolds of four dimensions, Bulletin of the American Mathematical Society, 51, 12, 1945, 964–971, http://www.ams.org/journals/bull/1945-51-12/S0002-9904-1945-08483-3/S0002-9904-1945-08483-3.pdf]

“It [Vatican II] does not look upon the ‘religious’ as one dimension among other dimensions of human existence. The religious dimension intersects with other dimensions. That is why the Council could speak of ‘the supremely human character’ of the Church’s religious mission.”

Kurien Kunnumpuram (1931–2018) Indian theologian

Kunnumpuram, K. (2005) Towards a New Humanity: Reflections on the Church's Mission in India Today. Mumbai: St Pauls
On the Church

“THEOREM: if G is a locally euclidean, connected, simply connected topological group of dimension n greater than one, then G contains a closed proper subgroup of positive dimension.”

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Deane Montgomery2

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