Minimal torus

On the other hand, given a torus with non-trivial translation group, there exists a unique minimal torus with the same universal cover and trivial translation subgroup. Those minimal tori correspond, in a one-to-one way, to periodic tilings of the plane. 

Planes 6, 12, 14, 20-23 (3-valent ones) and 27, 28, 30-32 are unique those 12 sporadic cases are represented on Figures 9.5 and 9.12, respectively. The notation (v, pa, pb) (under picture of plane) denotes the number of vertices, the number of a-gons and the number of fe-gons in a minimal torus obtained from the plane. 

When presenting a ( a, b, /c)-sphere in a drawing, we will indicate its number of vertices and its Schoenflies symmetry group (see Section 1.4). The notation (v, pa, pb) under picture of a minimal ( a, b], fc)-torus indicate its number of vertices, its number of a- and fo-gonal faces we also indicate its wallpaper symmetry group (see Section 1.4). 

Remark that in this formula e is given, being related to the size of the Q-neighborhood of the torus, but r is arbitrary. Therefore we may look for a best choice of r as a function of e so that the quantity (r + 1) T+1 er+l is minimized. It is an easy matter to check that the minimum is reached for... 

If we assume that the integral f on Q has more than one minimal and maximal circles, the assertion concerning infinity of the group Hi Q,Z) becomes invalid. We will give an example lens spaces obtained by gluing two full tori along the boundary torus. On such spaces one may set a function with exactly two critical circles, minimal and maximal. 

This is what Lacan aims at in his persistent reference to the torus and other variations of Mobius-strip-like structures in which the relationship between inside and outside is inverted if we want to grasp the minimal structure of subjectivity, the clear-cut opposition between inner subjective experience and outer objective reality is not sufficient—there is an excess on both sides. On the one hand, we should accept the lesson of Kant s transcendental idealism out of the confused multitude of impressions, objective reality emerges through the intervention of the subject s transcendental act. In other words, Kant does not deny the distinction between the multitude of subjective impressions and objective reality his point is merely that this very distinction results from the intervention of a subjective gesture of transcendental constitution. In a homologous way, Lacan s Master Signifier is the subjective signifying feature which sustains the very objective symbolic structure if we ab-... 

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