{\displaystyle T=\left\{(x,y,z,w)\in \mathbf {S} ^{3}\mid x^{2}+y^{2}={\frac {1}{2}},\ z^{2}+w^{2}={\frac {1}{2}}\right\}.} Other tori in S3 having this partitioning property include the square tori of the form Q⋅T, where Q is a rotation of 4-dimensional space R4, or in other words Q is a member of the Lie group SO(4). It is known that there exists no C2 (twice continuously differentiable) embedding of a flat torus into 3-space. (The idea of the proof is to take a large sphere containing such a flat torus in its interior, and shrink the radius of the sphere until it just touches the torus for the first time. Such a point of contact must be a tangency. But that would imply that part of the torus, since it has zero curvature everywhere, must lie strictly outside the sphere, which is a contradiction.) On the other hand, according to the Nash-Kuiper theorem, which was proven in the 1950s, an isometric C1 embedding exists. This is solely an existence proof and does not provide explicit equations for such an embedding. In April 2012, an explicit C1 (continuously differentiable) embedding of a flat torus into 3-dimensional Euclidean space R3 was found. It is similar in structure to a fractal as it is constructed by repeatedly corrugating an ordinary torus. Like fractals, it has no defined Gaussian curvature. However, unlike fractals, it does have defined surface normals. It is a flat torus in the sense that as metric spaces, it is isometric to a flat square torus.
Ludwigs angina (lat. : Angina ludovici) is a type of severe cellulitis involving the floor of the mouth and is often caused by bacterial sources. Early in the infection, floor of the mouth raises due to swelling, leading to difficulty swallowing saliva. As a result, patients may present with difficulty speaking and drooling. As the condition worsens, the airway may be compromised and hardening of the spaces on both sides of the tongue may develop. Overall, this condition has a rapid onset over a few hours. The majority of cases follow a dental infection. Other causes include a parapharyngeal abscess, mandibular fracture, cut or piercing inside the mouth, or submandibular salivary stones. The infection spreads through the connective tissue of the floor of the mouth and is normally caused by infectious and invasive organisms such as Streptococcus, Staphylococcus, and Bacteroides.Prevention is by appropriate dental care including management of dental infections. Initial treatment is generally with broad-spectrum antibiotics and corticosteroids. In more advanced cases endotracheal intubation or tracheostomy may be required.With the advent of antibiotics in 1940s, improved oral and dental hygiene, and more aggressive surgical approaches for treatment, the risk of death due to Ludwigs angina has significantly reduced. It is named after a German physician, Wilhelm Frederick von Ludwig, who first described this condition in 1836. Signs and symptoms
Ludwigs angina is a form of severe, widespread cellulitis of the floor of the mouth, usually with bilateral involvement.