The vast majority of currently observed geometric shapes of celestial bodies can be explained by a simple symmetry idea: the initial distribution of matter is invariant with respect to shifts, rotations, and scaling, but this distribution is unstable, so we have spontaneous symmetry breaking. According to statistical physics, among all possible transitions, the most probable are the ones that retain the largest number of symmetries. This explains the currently observed shapes and -- on the qualitative level -- their relative frequency. According to this idea, the most probable first transition is into a planar (pancake) shape, then into a logarithmic spiral, and other shapes like a straight line fragment (pickle) are less probable. This is exactly what we have observed until recently, but recent observations have shown that, in contrast to the currently observed galaxies, early galaxies are mostly pickle-shaped. In this paper, we provide a possible geometric explanation for this phenomenon: namely, according to modern physics, the proper space was originally more than 3-dimensional; later, the additional dimensions compactified and thus, became not directly observable. For galaxies formed at the time when the overall spatial dimension was 5 or larger, the pickle shape is indeed more symmetric than the planar shape -- and should, therefore, be prevailing -- exactly as what we observe.