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In this paper we prove that a smooth embedded paper Moebius band must have aspect ratio greater than $\sqrt 3$. We also prove that any sequence of smooth embedded paper Moebius bands whose aspect ratio converges to $\sqrt 3$ must converge, up to isometry, to an equilateral triangle of semi-perimeter $\sqrt 3$. These results resolve the optimal pape...
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Citations
... When in its ground state, Reference [5] proposes that proton charge and mass are coupled and continually regenerate each other. For each proton charge arc, this coupling may be represented in the form of a virtual optimal Möbius band [5,[19][20][21][22]. Reference [5] proposes that this implies the geometry of a GSQV proton is optimal, which may explain why free or chemically bound protons do not decay. ...
A nested surface vortex structure may be used to explain several properties of free or chemically bound protons. The circular Unruh and zitterbewegung effects are combined to show that it is plausible for the mass of an unobserved ground-state proton to exist on a spherical surface. Such a model is consistent with general relativity. The charge of an unobserved ground-state proton is assumed to exist on two massless oppositely charged shells well outside that of its mass sphere. These two charge shells are assumed to exist on the two surfaces of a spindle torus. This spindle torus structure offers geometric explanations for proton isospin, g-factor, and charge radius. This geometric model involves mathematics typically encountered by undergraduate physics and chemistry students. Upon interaction with other particles, this ground-state proton model transforms into the valence quarks, gluon flux tubes, and initial sea quarks of the standard quantum chromodynamics model.
