Problèmes anciens et nouveaux sur les polyèdres euclidienseuclidiens
http://www.math.univ-toulouse.fr/schlenker
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The discovery of the dodecahedron
The simplest regular polyhedra have been known forever ( ?) : the tetrahedron, cube, octahedron. According to later sources, the dodecahedron was discovered by a Pythagorean philosopher, Hippasus of Metapontum, born approx. 500bc. He is also credited with the discovery of irrational numbers.
According to Iamblichus, Hippasus was also the rst to draw or construct the 'sphere consisting of 12 regular pentagons' [...] and to make this construction public, which was considered a criminal divulgation of Pythagorean secret knowledge.
The discovery of incommensurability by Hippasus of Metapontum,
Kurt von Fritz, Annals of Math. 46 :2 (1945) pp. 242-264.
Hippasus is said to have been drowned for this publication.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The discovery of the dodecahedron
The simplest regular polyhedra have been known forever ( ?) : the tetrahedron, cube, octahedron. According to later sources, the dodecahedron was discovered by a Pythagorean philosopher, Hippasus of Metapontum, born approx. 500bc. He is also credited with the discovery of irrational numbers.
According to Iamblichus, Hippasus was also the rst to draw or construct the 'sphere consisting of 12 regular pentagons' [...] and to make this construction public, which was considered a criminal divulgation of Pythagorean secret knowledge.
The discovery of incommensurability by Hippasus of Metapontum,
Kurt von Fritz, Annals of Math. 46 :2 (1945) pp. 242-264.
Hippasus is said to have been drowned for this publication.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The discovery of the dodecahedron
The simplest regular polyhedra have been known forever ( ?) : the tetrahedron, cube, octahedron. According to later sources, the dodecahedron was discovered by a Pythagorean philosopher, Hippasus of Metapontum, born approx. 500bc. He is also credited with the discovery of irrational numbers.
According to Iamblichus, Hippasus was also the rst to draw or construct the 'sphere consisting of 12 regular pentagons' [...] and to make this construction public, which was considered a criminal divulgation of Pythagorean secret knowledge.
The discovery of incommensurability by Hippasus of Metapontum,
Kurt von Fritz, Annals of Math. 46 :2 (1945) pp. 242-264.
Hippasus is said to have been drowned for this publication.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The discovery of the dodecahedron
The simplest regular polyhedra have been known forever ( ?) : the tetrahedron, cube, octahedron. According to later sources, the dodecahedron was discovered by a Pythagorean philosopher, Hippasus of Metapontum, born approx. 500bc. He is also credited with the discovery of irrational numbers.
According to Iamblichus, Hippasus was also the rst to draw or construct the 'sphere consisting of 12 regular pentagons' [...] and to make this construction public, which was considered a criminal divulgation of Pythagorean secret knowledge.
The discovery of incommensurability by Hippasus of Metapontum,
Kurt von Fritz, Annals of Math. 46 :2 (1945) pp. 242-264.
Hippasus is said to have been drowned for this publication.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The discovery of the dodecahedron
The simplest regular polyhedra have been known forever ( ?) : the tetrahedron, cube, octahedron. According to later sources, the dodecahedron was discovered by a Pythagorean philosopher, Hippasus of Metapontum, born approx. 500bc. He is also credited with the discovery of irrational numbers.
According to Iamblichus, Hippasus was also the rst to draw or construct the 'sphere consisting of 12 regular pentagons' [...] and to make this construction public, which was considered a criminal divulgation of Pythagorean secret knowledge.
The discovery of incommensurability by Hippasus of Metapontum,
Kurt von Fritz, Annals of Math. 46 :2 (1945) pp. 242-264.
Hippasus is said to have been drowned for this publication.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The rst classication
Once the dodecahedron is known, the icosahedron is obtained by duality. The nal result of Euclid's Elements is the classication of regular polyhedra. Regular polyhedra had a huge inuence on greek philosophy.
According to Plato (427-347 BC), they were in relation to the four elements : the cube, with the earth, the tetrahedron, with re, the octahedron, with air, the icosahedron, with water. The dodecahedron serves for the nal arrang- ment of the universe ( ? ?) (Timeus)
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The rst classication
Once the dodecahedron is known, the icosahedron is obtained by duality. The nal result of Euclid's Elements is the classication of regular polyhedra. Regular polyhedra had a huge inuence on greek philosophy.
According to Plato (427-347 BC), they were in relation to the four elements : the cube, with the earth, the tetrahedron, with re, the octahedron, with air, the icosahedron, with water. The dodecahedron serves for the nal arrang- ment of the universe ( ? ?) (Timeus)
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The rst classication
Once the dodecahedron is known, the icosahedron is obtained by duality. The nal result of Euclid's Elements is the classication of regular polyhedra. Regular polyhedra had a huge inuence on greek philosophy.
According to Plato (427-347 BC), they were in relation to the four elements : the cube, with the earth, the tetrahedron, with re, the octahedron, with air, the icosahedron, with water. The dodecahedron serves for the nal arrang- ment of the universe ( ? ?) (Timeus)
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The rst classication
Once the dodecahedron is known, the icosahedron is obtained by duality. The nal result of Euclid's Elements is the classication of regular polyhedra. Regular polyhedra had a huge inuence on greek philosophy.
According to Plato (427-347 BC), they were in relation to the four elements : the cube, with the earth, the tetrahedron, with re, the octahedron, with air, the icosahedron, with water. The dodecahedron serves for the nal arrang- ment of the universe ( ? ?) (Timeus)
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The rst classication
Once the dodecahedron is known, the icosahedron is obtained by duality. The nal result of Euclid's Elements is the classication of regular polyhedra. Regular polyhedra had a huge inuence on greek philosophy.
According to Plato (427-347 BC), they were in relation to the four elements : the cube, with the earth, the tetrahedron, with re, the octahedron, with air, the icosahedron, with water. The dodecahedron serves for the nal arrang- ment of the universe ( ? ?) (Timeus)
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The rst classication
Once the dodecahedron is known, the icosahedron is obtained by duality. The nal result of Euclid's Elements is the classication of regular polyhedra. Regular polyhedra had a huge inuence on greek philosophy.
According to Plato (427-347 BC), they were in relation to the four elements : the cube, with the earth, the tetrahedron, with re, the octahedron, with air, the icosahedron, with water. The dodecahedron serves for the nal arrang- ment of the universe ( ? ?) (Timeus)
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The rst classication
Once the dodecahedron is known, the icosahedron is obtained by duality. The nal result of Euclid's Elements is the classication of regular polyhedra. Regular polyhedra had a huge inuence on greek philosophy.
According to Plato (427-347 BC), they were in relation to the four elements : the cube, with the earth, the tetrahedron, with re, the octahedron, with air, the icosahedron, with water. The dodecahedron serves for the nal arrang- ment of the universe ( ? ?) (Timeus)
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The rst classication
Once the dodecahedron is known, the icosahedron is obtained by duality. The nal result of Euclid's Elements is the classication of regular polyhedra. Regular polyhedra had a huge inuence on greek philosophy.
According to Plato (427-347 BC), they were in relation to the four elements : the cube, with the earth, the tetrahedron, with re, the octahedron, with air, the icosahedron, with water. The dodecahedron serves for the nal arrang- ment of the universe ( ? ?) (Timeus)
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The rst classication
Once the dodecahedron is known, the icosahedron is obtained by duality. The nal result of Euclid's Elements is the classication of regular polyhedra. Regular polyhedra had a huge inuence on greek philosophy.
According to Plato (427-347 BC), they were in relation to the four elements : the cube, with the earth, the tetrahedron, with re, the octahedron, with air, the icosahedron, with water. The dodecahedron serves for the nal arrang- ment of the universe ( ? ?) (Timeus)
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Semi-regular polyhedra
The classica- tion was extended by Archimedes (287-212 BC) to semi-regular polyhedra :
2 innite families, the prisms and anti-prisms, and 13 sporadic polyhedra. However his work was lost.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Semi-regular polyhedra
The classica- tion was extended by Archimedes (287-212 BC) to semi-regular polyhedra :
2 innite families, the prisms and anti-prisms, and 13 sporadic polyhedra. However his work was lost.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Semi-regular polyhedra
The classica- tion was extended by Archimedes (287-212 BC) to semi-regular polyhedra :
2 innite families, the prisms and anti-prisms, and 13 sporadic polyhedra. However his work was lost.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Semi-regular polyhedra
The classica- tion was extended by Archimedes (287-212 BC) to semi-regular polyhedra :
2 innite families, the prisms and anti-prisms, and 13 sporadic polyhedra. However his work was lost.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Painters and polyhedra
A strong interest for polyhedra appears during the Renaissance, with painters like Albrecht Dürer (1471-1528). They often appear in drawings and archi- tectural motives, for instance in the work of Luca Pacioli (1445-1517), illustrated by Leonardo da Vinci. Dürer also introduces a new description of polyhedra, in a at form. The Archi- medean polyhedra are re-discovered.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Painters and polyhedra
A strong interest for polyhedra appears during the Renaissance, with painters like Albrecht Dürer (1471-1528). They often appear in drawings and archi- tectural motives, for instance in the work of Luca Pacioli (1445-1517), illustrated by Leonardo da Vinci. Dürer also introduces a new description of polyhedra, in a at form. The Archi- medean polyhedra are re-discovered.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Painters and polyhedra
A strong interest for polyhedra appears during the Renaissance, with painters like Albrecht Dürer (1471-1528). They often appear in drawings and archi- tectural motives, for instance in the work of Luca Pacioli (1445-1517), illustrated by Leonardo da Vinci. Dürer also introduces a new description of polyhedra, in a at form. The Archi- medean polyhedra are re-discovered.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Painters and polyhedra
A strong interest for polyhedra appears during the Renaissance, with painters like Albrecht Dürer (1471-1528). They often appear in drawings and archi- tectural motives, for instance in the work of Luca Pacioli (1445-1517), illustrated by Leonardo da Vinci. Dürer also introduces a new description of polyhedra, in a at form. The Archi- medean polyhedra are re-discovered.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Painters and polyhedra
A strong interest for polyhedra appears during the Renaissance, with painters like Albrecht Dürer (1471-1528). They often appear in drawings and archi- tectural motives, for instance in the work of Luca Pacioli (1445-1517), illustrated by Leonardo da Vinci. Dürer also introduces a new description of polyhedra, in a at form. The Archi- medean polyhedra are re-discovered.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Painters and polyhedra
A strong interest for polyhedra appears during the Renaissance, with painters like Albrecht Dürer (1471-1528). They often appear in drawings and archi- tectural motives, for instance in the work of Luca Pacioli (1445-1517), illustrated by Leonardo da Vinci. Dürer also introduces a new description of polyhedra, in a at form. The Archi- medean polyhedra are re-discovered.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Painters and polyhedra
A strong interest for polyhedra appears during the Renaissance, with painters like Albrecht Dürer (1471-1528). They often appear in drawings and archi- tectural motives, for instance in the work of Luca Pacioli (1445-1517), illustrated by Leonardo da Vinci. Dürer also introduces a new description of polyhedra, in a at form. The Archi- medean polyhedra are re-discovered.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Painters and polyhedra
A strong interest for polyhedra appears during the Renaissance, with painters like Albrecht Dürer (1471-1528). They often appear in drawings and archi- tectural motives, for instance in the work of Luca Pacioli (1445-1517), illustrated by Leonardo da Vinci. Dürer also introduces a new description of polyhedra, in a at form. The Archi- medean polyhedra are re-discovered.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Kepler and the non-convex polyhedra
Johannes Kepler (1571-1630) was fasci- nated by greek mathematics (polyhedra and conics). Kepler rst recovers Archi- medes' classication of semi-regular po- lyhedra. He then realizes that Euclid only considers, implicitly, convex polyhedra, and discovers two new, non-convex regu- lar polyhedra. The list will be completed by Poinsot (1777-1859), who recovers Kepler's two polyhedra and discovers two other ones. The classication of Euclidean polyhedra is nished, to millenia after Euclid.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Kepler and the non-convex polyhedra
Johannes Kepler (1571-1630) was fasci- nated by greek mathematics (polyhedra and conics). Kepler rst recovers Archi- medes' classication of semi-regular po- lyhedra. He then realizes that Euclid only considers, implicitly, convex polyhedra, and discovers two new, non-convex regu- lar polyhedra. The list will be completed by Poinsot (1777-1859), who recovers Kepler's two polyhedra and discovers two other ones. The classication of Euclidean polyhedra is nished, to millenia after Euclid.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Kepler and the non-convex polyhedra
Johannes Kepler (1571-1630) was fasci- nated by greek mathematics (polyhedra and conics). Kepler rst recovers Archi- medes' classication of semi-regular po- lyhedra. He then realizes that Euclid only considers, implicitly, convex polyhedra, and discovers two new, non-convex regu- lar polyhedra. The list will be completed by Poinsot (1777-1859), who recovers Kepler's two polyhedra and discovers two other ones. The classication of Euclidean polyhedra is nished, to millenia after Euclid.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Kepler and the non-convex polyhedra
Johannes Kepler (1571-1630) was fasci- nated by greek mathematics (polyhedra and conics). Kepler rst recovers Archi- medes' classication of semi-regular po- lyhedra. He then realizes that Euclid only considers, implicitly, convex polyhedra, and discovers two new, non-convex regu- lar polyhedra. The list will be completed by Poinsot (1777-1859), who recovers Kepler's two polyhedra and discovers two other ones. The classication of Euclidean polyhedra is nished, to millenia after Euclid.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Kepler and the non-convex polyhedra
Johannes Kepler (1571-1630) was fasci- nated by greek mathematics (polyhedra and conics). Kepler rst recovers Archi- medes' classication of semi-regular po- lyhedra. He then realizes that Euclid only considers, implicitly, convex polyhedra, and discovers two new, non-convex regu- lar polyhedra. The list will be completed by Poinsot (1777-1859), who recovers Kepler's two polyhedra and discovers two other ones. The classication of Euclidean polyhedra is nished, to millenia after Euclid.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Kepler and the non-convex polyhedra
Johannes Kepler (1571-1630) was fasci- nated by greek mathematics (polyhedra and conics). Kepler rst recovers Archi- medes' classication of semi-regular po- lyhedra. He then realizes that Euclid only considers, implicitly, convex polyhedra, and discovers two new, non-convex regu- lar polyhedra. The list will be completed by Poinsot (1777-1859), who recovers Kepler's two polyhedra and discovers two other ones. The classication of Euclidean polyhedra is nished, to millenia after Euclid.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Kepler and the non-convex polyhedra
Johannes Kepler (1571-1630) was fasci- nated by greek mathematics (polyhedra and conics). Kepler rst recovers Archi- medes' classication of semi-regular po- lyhedra. He then realizes that Euclid only considers, implicitly, convex polyhedra, and discovers two new, non-convex regu- lar polyhedra. The list will be completed by Poinsot (1777-1859), who recovers Kepler's two polyhedra and discovers two other ones. The classication of Euclidean polyhedra is nished, to millenia after Euclid.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Kepler and the non-convex polyhedra
Johannes Kepler (1571-1630) was fasci- nated by greek mathematics (polyhedra and conics). Kepler rst recovers Archi- medes' classication of semi-regular po- lyhedra. He then realizes that Euclid only considers, implicitly, convex polyhedra, and discovers two new, non-convex regu- lar polyhedra. The list will be completed by Poinsot (1777-1859), who recovers Kepler's two polyhedra and discovers two other ones. The classication of Euclidean polyhedra is nished, to millenia after Euclid.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The rigidity of polyhedra
Another branch of the geometry of po- lyhedra appears when Legendre reads a (bad) translation of Euclid's Elements. Legendre discovers that two polyhedra with isometric corresponding faces are congruent. The proof is written only in 1813 by Cauchy (1789-1857), with two gaps. Applications : rigidity of convex surfaces, Alexandrov rigidity, isometric realization for smooth and polyhedral metrics...
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The rigidity of polyhedra
Another branch of the geometry of po- lyhedra appears when Legendre reads a (bad) translation of Euclid's Elements. Legendre discovers that two polyhedra with isometric corresponding faces are congruent. The proof is written only in 1813 by Cauchy (1789-1857), with two gaps. Applications : rigidity of convex surfaces, Alexandrov rigidity, isometric realization for smooth and polyhedral metrics...
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The rigidity of polyhedra
Another branch of the geometry of po- lyhedra appears when Legendre reads a (bad) translation of Euclid's Elements. Legendre discovers that two polyhedra with isometric corresponding faces are congruent. The proof is written only in 1813 by Cauchy (1789-1857), with two gaps. Applications : rigidity of convex surfaces, Alexandrov rigidity, isometric realization for smooth and polyhedral metrics...
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The rigidity of polyhedra
Another branch of the geometry of po- lyhedra appears when Legendre reads a (bad) translation of Euclid's Elements. Legendre discovers that two polyhedra with isometric corresponding faces are congruent. The proof is written only in 1813 by Cauchy (1789-1857), with two gaps. Applications : rigidity of convex surfaces, Alexandrov rigidity, isometric realization for smooth and polyhedral metrics...
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Innitesimal rigidity
A polyhedron is inf. rigid if inf. deforma- tions lead to inf. variation of the sha- pes of the faces. Important for engi- neers/architects. Dehn (1916) proves that convex polyhe- dra are inf. rigid. A new question came up : what about non-convex polyhedra ? They can be inf. exible and even exible (Connelly 1978).
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Innitesimal rigidity
A polyhedron is inf. rigid if inf. deforma- tions lead to inf. variation of the sha- pes of the faces. Important for engi- neers/architects. Dehn (1916) proves that convex polyhe- dra are inf. rigid. A new question came up : what about non-convex polyhedra ? They can be inf. exible and even exible (Connelly 1978).
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Innitesimal rigidity
A polyhedron is inf. rigid if inf. deforma- tions lead to inf. variation of the sha- pes of the faces. Important for engi- neers/architects. Dehn (1916) proves that convex polyhe- dra are inf. rigid. A new question came up : what about non-convex polyhedra ? They can be inf. exible and even exible (Connelly 1978).
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Innitesimal rigidity
A polyhedron is inf. rigid if inf. deforma- tions lead to inf. variation of the sha- pes of the faces. Important for engi- neers/architects. Dehn (1916) proves that convex polyhe- dra are inf. rigid. A new question came up : what about non-convex polyhedra ? They can be inf. exible and even exible (Connelly 1978).
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Innitesimal rigidity
A polyhedron is inf. rigid if inf. deforma- tions lead to inf. variation of the sha- pes of the faces. Important for engi- neers/architects. Dehn (1916) proves that convex polyhe- dra are inf. rigid. A new question came up : what about non-convex polyhedra ? They can be inf. exible and even exible (Connelly 1978).
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Hyperbolic polyhedra and dihedral angles
Questions on the geometry of polyhedra can be considered in the hyperbolic/spherical context. The inf. rigidity of polyhedra turns out to be a projective property (Darboux, Sauer), independent of the geometry (Pogorelov). For hyperbolic polyhedra dihedral angles are more important than edge lengths, due to Poincaré's theorem : the group generated by reections in the faces is discrete i the dihedral angles are π/k. Andreev (1970) characterized the possible dihedral angles (< π/2) of nite volume hyperbolic polyhedra. Rivin and Hodgson (1994) showed that this is a consequence of an isometric realization theorem in a space dual to the hyperbolic space (Cauchy's thm is important). Spaces obtained from polyhedra can even be found in the real world. For instance the universe might be a dodecahedral space.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Hyperbolic polyhedra and dihedral angles
Questions on the geometry of polyhedra can be considered in the hyperbolic/spherical context. The inf. rigidity of polyhedra turns out to be a projective property (Darboux, Sauer), independent of the geometry (Pogorelov). For hyperbolic polyhedra dihedral angles are more important than edge lengths, due to Poincaré's theorem : the group generated by reections in the faces is discrete i the dihedral angles are π/k. Andreev (1970) characterized the possible dihedral angles (< π/2) of nite volume hyperbolic polyhedra. Rivin and Hodgson (1994) showed that this is a consequence of an isometric realization theorem in a space dual to the hyperbolic space (Cauchy's thm is important). Spaces obtained from polyhedra can even be found in the real world. For instance the universe might be a dodecahedral space.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Hyperbolic polyhedra and dihedral angles
Questions on the geometry of polyhedra can be considered in the hyperbolic/spherical context. The inf. rigidity of polyhedra turns out to be a projective property (Darboux, Sauer), independent of the geometry (Pogorelov). For hyperbolic polyhedra dihedral angles are more important than edge lengths, due to Poincaré's theorem : the group generated by reections in the faces is discrete i the dihedral angles are π/k. Andreev (1970) characterized the possible dihedral angles (< π/2) of nite volume hyperbolic polyhedra. Rivin and Hodgson (1994) showed that this is a consequence of an isometric realization theorem in a space dual to the hyperbolic space (Cauchy's thm is important). Spaces obtained from polyhedra can even be found in the real world. For instance the universe might be a dodecahedral space.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Hyperbolic polyhedra and dihedral angles
Questions on the geometry of polyhedra can be considered in the hyperbolic/spherical context. The inf. rigidity of polyhedra turns out to be a projective property (Darboux, Sauer), independent of the geometry (Pogorelov). For hyperbolic polyhedra dihedral angles are more important than edge lengths, due to Poincaré's theorem : the group generated by reections in the faces is discrete i the dihedral angles are π/k. Andreev (1970) characterized the possible dihedral angles (< π/2) of nite volume hyperbolic polyhedra. Rivin and Hodgson (1994) showed that this is a consequence of an isometric realization theorem in a space dual to the hyperbolic space (Cauchy's thm is important). Spaces obtained from polyhedra can even be found in the real world. For instance the universe might be a dodecahedral space.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Hyperbolic polyhedra and dihedral angles
Questions on the geometry of polyhedra can be considered in the hyperbolic/spherical context. The inf. rigidity of polyhedra turns out to be a projective property (Darboux, Sauer), independent of the geometry (Pogorelov). For hyperbolic polyhedra dihedral angles are more important than edge lengths, due to Poincaré's theorem : the group generated by reections in the faces is discrete i the dihedral angles are π/k. Andreev (1970) characterized the possible dihedral angles (< π/2) of nite volume hyperbolic polyhedra. Rivin and Hodgson (1994) showed that this is a consequence of an isometric realization theorem in a space dual to the hyperbolic space (Cauchy's thm is important). Spaces obtained from polyhedra can even be found in the real world. For instance the universe might be a dodecahedral space.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Hyperbolic polyhedra and dihedral angles
Questions on the geometry of polyhedra can be considered in the hyperbolic/spherical context. The inf. rigidity of polyhedra turns out to be a projective property (Darboux, Sauer), independent of the geometry (Pogorelov). For hyperbolic polyhedra dihedral angles are more important than edge lengths, due to Poincaré's theorem : the group generated by reections in the faces is discrete i the dihedral angles are π/k. Andreev (1970) characterized the possible dihedral angles (< π/2) of nite volume hyperbolic polyhedra. Rivin and Hodgson (1994) showed that this is a consequence of an isometric realization theorem in a space dual to the hyperbolic space (Cauchy's thm is important). Spaces obtained from polyhedra can even be found in the real world. For instance the universe might be a dodecahedral space.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Application in physics
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Rigidity wrt dihedral angles
The Andreev thm suggests the following : are hyperbolic polyhedra inf rigid with respect to their dihedral angles ? Yes (Andreev) for acute angles. Analog for Euclidean polyhedra : does the dihedral angles determine the face angles (Stoker) ? Answer : Yes, recently announced by Mazzeo & Montcouquiol. Proof uses dicult analytic arguments (elliptic operators on cone-mds), no elementary proof known.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Rigidity wrt dihedral angles
The Andreev thm suggests the following : are hyperbolic polyhedra inf rigid with respect to their dihedral angles ? Yes (Andreev) for acute angles. Analog for Euclidean polyhedra : does the dihedral angles determine the face angles (Stoker) ? Answer : Yes, recently announced by Mazzeo & Montcouquiol. Proof uses dicult analytic arguments (elliptic operators on cone-mds), no elementary proof known.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Rigidity wrt dihedral angles
The Andreev thm suggests the following : are hyperbolic polyhedra inf rigid with respect to their dihedral angles ? Yes (Andreev) for acute angles. Analog for Euclidean polyhedra : does the dihedral angles determine the face angles (Stoker) ? Answer : Yes, recently announced by Mazzeo & Montcouquiol. Proof uses dicult analytic arguments (elliptic operators on cone-mds), no elementary proof known.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Rigidity wrt dihedral angles
The Andreev thm suggests the following : are hyperbolic polyhedra inf rigid with respect to their dihedral angles ? Yes (Andreev) for acute angles. Analog for Euclidean polyhedra : does the dihedral angles determine the face angles (Stoker) ? Answer : Yes, recently announced by Mazzeo & Montcouquiol. Proof uses dicult analytic arguments (elliptic operators on cone-mds), no elementary proof known.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Unfolding
Recall Dürer's at representation of a polyhedron. Q : is it always possible to unfold a polyhedron in this way ? Cut only along edges (otherwise, easy). Q : let g be a at metric with cone singularities on S2, angles < 2π, with a decomposition in convex polygons. Can one unfold by cutting along some of the edges ?
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Unfolding
Recall Dürer's at representation of a polyhedron. Q : is it always possible to unfold a polyhedron in this way ? Cut only along edges (otherwise, easy). Q : let g be a at metric with cone singularities on S2, angles < 2π, with a decomposition in convex polygons. Can one unfold by cutting along some of the edges ?
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Unfolding
Recall Dürer's at representation of a polyhedron. Q : is it always possible to unfold a polyhedron in this way ? Cut only along edges (otherwise, easy). Q : let g be a at metric with cone singularities on S2, angles < 2π, with a decomposition in convex polygons. Can one unfold by cutting along some of the edges ?
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Unfolding
Recall Dürer's at representation of a polyhedron. Q : is it always possible to unfold a polyhedron in this way ? Cut only along edges (otherwise, easy). Q : let g be a at metric with cone singularities on S2, angles < 2π, with a decomposition in convex polygons. Can one unfold by cutting along some of the edges ?
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Rigidity of weakly convex polyhedra
Def : P ⊂ R3 is weakly convex if its vertices are the vertices of a (strictly) convex polyhedron. Def. P is decomposable if it can be cut into convex pieces without adding an interior vertex. Conj. A weakly convex, decomposable polyhedron is inf. rigid. Extension of the Cauchy-Dehn result. The hypothesis that P is decomposable is necessary : ∃ a (non-decomposable) octahedron, with vertices on S2, not inf. rigid.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Rigidity of weakly convex polyhedra
Def : P ⊂ R3 is weakly convex if its vertices are the vertices of a (strictly) convex polyhedron. Def. P is decomposable if it can be cut into convex pieces without adding an interior vertex. Conj. A weakly convex, decomposable polyhedron is inf. rigid. Extension of the Cauchy-Dehn result. The hypothesis that P is decomposable is necessary : ∃ a (non-decomposable) octahedron, with vertices on S2, not inf. rigid.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Rigidity of weakly convex polyhedra
Def : P ⊂ R3 is weakly convex if its vertices are the vertices of a (strictly) convex polyhedron. Def. P is decomposable if it can be cut into convex pieces without adding an interior vertex. Conj. A weakly convex, decomposable polyhedron is inf. rigid. Extension of the Cauchy-Dehn result. The hypothesis that P is decomposable is necessary : ∃ a (non-decomposable) octahedron, with vertices on S2, not inf. rigid.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Rigidity of weakly convex polyhedra
Def : P ⊂ R3 is weakly convex if its vertices are the vertices of a (strictly) convex polyhedron. Def. P is decomposable if it can be cut into convex pieces without adding an interior vertex. Conj. A weakly convex, decomposable polyhedron is inf. rigid. Extension of the Cauchy-Dehn result. The hypothesis that P is decomposable is necessary : ∃ a (non-decomposable) octahedron, with vertices on S2, not inf. rigid.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Rigidity of weakly convex polyhedra
Def : P ⊂ R3 is weakly convex if its vertices are the vertices of a (strictly) convex polyhedron. Def. P is decomposable if it can be cut into convex pieces without adding an interior vertex. Conj. A weakly convex, decomposable polyhedron is inf. rigid. Extension of the Cauchy-Dehn result. The hypothesis that P is decomposable is necessary : ∃ a (non-decomposable) octahedron, with vertices on S2, not inf. rigid.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
A partial result
Def : P is codecomposable if one can add simplices so as to go from P to its convex hull through weakly convex polyhedra. Thm A (Izmestiev, S.) : any weakly convex, decomposable and codecomposable polyhedron, is innitesimally rigid. The hypothesis that P is codecomposable might not be necessary ( ? ?) It is very weak for polyhedra homeomorphic to a sphere, however an example of a weakly convex, non codecomposable polyhedron was found in 2002 (Aichholzer, Alboul, Hurtado). Some previous results, some in collaboration w/ Bob Connelly.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
A partial result
Def : P is codecomposable if one can add simplices so as to go from P to its convex hull through weakly convex polyhedra. Thm A (Izmestiev, S.) : any weakly convex, decomposable and codecomposable polyhedron, is innitesimally rigid. The hypothesis that P is codecomposable might not be necessary ( ? ?) It is very weak for polyhedra homeomorphic to a sphere, however an example of a weakly convex, non codecomposable polyhedron was found in 2002 (Aichholzer, Alboul, Hurtado). Some previous results, some in collaboration w/ Bob Connelly.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
A partial result
Def : P is codecomposable if one can add simplices so as to go from P to its convex hull through weakly convex polyhedra. Thm A (Izmestiev, S.) : any weakly convex, decomposable and codecomposable polyhedron, is innitesimally rigid. The hypothesis that P is codecomposable might not be necessary ( ? ?) It is very weak for polyhedra homeomorphic to a sphere, however an example of a weakly convex, non codecomposable polyhedron was found in 2002 (Aichholzer, Alboul, Hurtado). Some previous results, some in collaboration w/ Bob Connelly.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
A partial result
Def : P is codecomposable if one can add simplices so as to go from P to its convex hull through weakly convex polyhedra. Thm A (Izmestiev, S.) : any weakly convex, decomposable and codecomposable polyhedron, is innitesimally rigid. The hypothesis that P is codecomposable might not be necessary ( ? ?) It is very weak for polyhedra homeomorphic to a sphere, however an example of a weakly convex, non codecomposable polyhedron was found in 2002 (Aichholzer, Alboul, Hurtado). Some previous results, some in collaboration w/ Bob Connelly.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
A partial result
Def : P is codecomposable if one can add simplices so as to go from P to its convex hull through weakly convex polyhedra. Thm A (Izmestiev, S.) : any weakly convex, decomposable and codecomposable polyhedron, is innitesimally rigid. The hypothesis that P is codecomposable might not be necessary ( ? ?) It is very weak for polyhedra homeomorphic to a sphere, however an example of a weakly convex, non codecomposable polyhedron was found in 2002 (Aichholzer, Alboul, Hurtado). Some previous results, some in collaboration w/ Bob Connelly.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Tools for the proof
The proof uses three types of notions/results, some old, some recent.
the Hilbert-Einstein functional
geometric combinatorics/algebraic geometry : a (dicult) result of Morelli and (independently) Wlodarczyk (1996).
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Tools for the proof
The proof uses three types of notions/results, some old, some recent.
the Hilbert-Einstein functional
geometric combinatorics/algebraic geometry : a (dicult) result of Morelli and (independently) Wlodarczyk (1996).
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Tools for the proof
The proof uses three types of notions/results, some old, some recent.
the Hilbert-Einstein functional
geometric combinatorics/algebraic geometry : a (dicult) result of Morelli and (independently) Wlodarczyk (1996).
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Tools for the proof
The proof uses three types of notions/results, some old, some recent.
the Hilbert-Einstein functional
geometric combinatorics/algebraic geometry : a (dicult) result of Morelli and (independently) Wlodarczyk (1996).
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The Hilbert-Einstein functional
Let (M, g) be a Riemannian md, with scalar curvature Scg . The Hilbert-Einstein functional :
S(g) =
∫ M
Scgdvg .
Thm (Hilbert ?) : S(M, g) is critical, among metrics of xed volume, i g is Einstein. With boundary : term in integral mean curvature. Thm (Blaschke, Herglotz) : S(M, ∂M, g) is critical, among metrics with xed induced metric on ∂M, i g is Einstein. Problem : S is neither convex nor concave hard to nd critical points. Not used much in practice, eg to nd Einstein metrics.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The Hilbert-Einstein functional
Let (M, g) be a Riemannian md, with scalar curvature Scg . The Hilbert-Einstein functional :
S(g) =
∫ M
Scgdvg .
Thm (Hilbert ?) : S(M, g) is critical, among metrics of xed volume, i g is Einstein. With boundary : term in integral mean curvature. Thm (Blaschke, Herglotz) : S(M, ∂M, g) is critical, among metrics with xed induced metric on ∂M, i g is Einstein. Problem : S is neither convex nor concave hard to nd critical points. Not used much in practice, eg to nd Einstein metrics.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The Hilbert-Einstein functional
Let (M, g) be a Riemannian md, with scalar curvature Scg . The Hilbert-Einstein functional :
S(g) =
∫ M
Scgdvg .
Thm (Hilbert ?) : S(M, g) is critical, among metrics of xed volume, i g is Einstein. With boundary : term in integral mean curvature. Thm (Blaschke, Herglotz) : S(M, ∂M, g) is critical, among metrics with xed induced metric on ∂M, i g is Einstein. Problem : S is neither convex nor concave hard to nd critical points. Not used much in practice, eg to nd Einstein metrics.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The Hilbert-Einstein functional
Let (M, g) be a Riemannian md, with scalar curvature Scg . The Hilbert-Einstein functional :
S(g) =
∫ M
Scgdvg .
Thm (Hilbert ?) : S(M, g) is critical, among metrics of xed volume, i g is Einstein. With boundary : term in integral mean curvature. Thm (Blaschke, Herglotz) : S(M, ∂M, g) is critical, among metrics with xed induced metric on ∂M, i g is Einstein. Problem : S is neither convex nor concave hard to nd critical points. Not used much in practice, eg to nd Einstein metrics.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The Hilbert-Einstein functional
Let (M, g) be a Riemannian md, with scalar curvature Scg . The Hilbert-Einstein functional :
S(g) =
∫ M
Scgdvg .
Thm (Hilbert ?) : S(M, g) is critical, among metrics of xed volume, i g is Einstein. With boundary : term in integral mean curvature. Thm (Blaschke, Herglotz) : S(M, ∂M, g) is critical, among metrics with xed induced metric on ∂M, i g is Einstein. Problem : S is neither convex nor concave hard to nd critical points. Not used much in practice, eg to nd Einstein metrics.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The Hilbert-Einstein functional
Let (M, g) be a Riemannian md, with scalar curvature Scg . The Hilbert-Einstein functional :
S(g) =
∫ M
Scgdvg .
Thm (Hilbert ?) : S(M, g) is critical, among metrics of xed volume, i g is Einstein. With boundary : term in integral mean curvature. Thm (Blaschke, Herglotz) : S(M, ∂M, g) is critical, among metrics with xed induced metric on ∂M, i g is Einstein. Problem : S is neither convex nor concave hard to nd critical points. Not used much in practice, eg to nd Einstein metrics.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The Schläi formula
ledθe = 0 .
Consider a polyhedron P with a triangulation T , and deformations which vary the length of the interior angles, xing the boundary metric. Cone singularities appear on the interior edges : total angle is not 2π. Def :
S = ∑ e∈Ei
∑ e∈Ei
(2π− θe)dle , and the Hessian of S is −MT = −(∂θi/∂lj). Remark : if T has no interior vertex, P is inf rigid i MT is non-degenerate.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The Schläi formula
ledθe = 0 .
Consider a polyhedron P with a triangulation T , and deformations which vary the length of the interior angles, xing the boundary metric. Cone singularities appear on the interior edges : total angle is not 2π. Def :
S = ∑ e∈Ei
∑ e∈Ei
(2π− θe)dle , and the Hessian of S is −MT = −(∂θi/∂lj). Remark : if T has no interior vertex, P is inf rigid i MT is non-degenerate.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The Schläi formula
ledθe = 0 .
Consider a polyhedron P with a triangulation T , and deformations which vary the length of the interior angles, xing the boundary metric. Cone singularities appear on the interior edges : total angle is not 2π. Def :
S = ∑ e∈Ei
∑ e∈Ei
(2π− θe)dle , and the Hessian of S is −MT = −(∂θi/∂lj). Remark : if T has no interior vertex, P is inf rigid i MT is non-degenerate.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The Schläi formula
ledθe = 0 .
Consider a polyhedron P with a triangulation T , and deformations which vary the length of the interior angles, xing the boundary metric. Cone singularities appear on the interior edges : total angle is not 2π. Def :
S = ∑ e∈Ei
∑ e∈Ei
(2π− θe)dle , and the Hessian of S is −MT = −(∂θi/∂lj). Remark : if T has no interior vertex, P is inf rigid i MT is non-degenerate.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The Schläi formula
ledθe = 0 .
Consider a polyhedron P with a triangulation T , and deformations which vary the length of the interior angles, xing the boundary metric. Cone singularities appear on the interior edges : total angle is not 2π. Def :
S = ∑ e∈Ei
∑ e∈Ei
(2π− θe)dle , and the Hessian of S is −MT = −(∂θi/∂lj). Remark : if T has no interior vertex, P is inf rigid i MT is non-degenerate.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The Schläi formula
ledθe = 0 .
Consider a polyhedron P with a triangulation T , and deformations which vary the length of the interior angles, xing the boundary metric. Cone singularities appear on the interior edges : total angle is not 2π. Def :
S = ∑ e∈Ei
∑ e∈Ei
(2π− θe)dle , and the Hessian of S is −MT = −(∂θi/∂lj). Remark : if T has no interior vertex, P is inf rigid i MT is non-degenerate.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The Schläi formula
ledθe = 0 .
Consider a polyhedron P with a triangulation T , and deformations which vary the length of the interior angles, xing the boundary metric. Cone singularities appear on the interior edges : total angle is not 2π. Def :
S = ∑ e∈Ei
∑ e∈Ei
(2π− θe)dle , and the Hessian of S is −MT = −(∂θi/∂lj). Remark : if T has no interior vertex, P is inf rigid i MT is non-degenerate.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The Schläi formula
ledθe = 0 .
Consider a polyhedron P with a triangulation T , and deformations which vary the length of the interior angles, xing the boundary metric. Cone singularities appear on the interior edges : total angle is not 2π. Def :
S = ∑ e∈Ei
∑ e∈Ei
(2π− θe)dle , and the Hessian of S is −MT = −(∂θi/∂lj). Remark : if T has no interior vertex, P is inf rigid i MT is non-degenerate.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
The Schläi formula
ledθe = 0 .
Consider a polyhedron P with a triangulation T , and deformations which vary the length of the interior angles, xing the boundary metric. Cone singularities appear on the interior edges : total angle is not 2π. Def :
S = ∑ e∈Ei
∑ e∈Ei
(2π− θe)dle , and the Hessian of S is −MT = −(∂θi/∂lj). Remark : if T has no interior vertex, P is inf rigid i MT is non-degenerate.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Concavity of S
Thm B. Let Q be a convex polyhedron, and let T be any triangulation without interior vertex. Then MT is positive denite.
Thm B =⇒ Thm A : start with a weakly convex, decomposable and codecomposable polyhedron P0. Choose a concave edge e1 and ll it with a simplex, to get P1 e1 is an interior edge of P1. Repeat with e2, etc. After a nite number of steps, Pn is the convex hull of P0. To go from Mk+1 to Mk , simply remove a line and a column (corresponding to ek+1). Since Mn is positive denite, so is M0. Therefore P0 is inf rigid, qed.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Concavity of S
Thm B. Let Q be a convex polyhedron, and let T be any triangulation without interior vertex. Then MT is positive denite.
Thm B =⇒ Thm A : start with a weakly convex, decomposable and codecomposable polyhedron P0. Choose a concave edge e1 and ll it with a simplex, to get P1 e1 is an interior edge of P1. Repeat with e2, etc. After a nite number of steps, Pn is the convex hull of P0. To go from Mk+1 to Mk , simply remove a line and a column (corresponding to ek+1). Since Mn is positive denite, so is M0. Therefore P0 is inf rigid, qed.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Concavity of S
Thm B. Let Q be a convex polyhedron, and let T be any triangulation without interior vertex. Then MT is positive denite.
Thm B =⇒ Thm A : start with a weakly convex, decomposable and codecomposable polyhedron P0. Choose a concave edge e1 and ll it with a simplex, to get P1 e1 is an interior edge of P1. Repeat with e2, etc. After a nite number of steps, Pn is the convex hull of P0. To go from Mk+1 to Mk , simply remove a line and a column (corresponding to ek+1). Since Mn is positive denite, so is M0. Therefore P0 is inf rigid, qed.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Concavity of S
Thm B. Let Q be a convex polyhedron, and let T be any triangulation without interior vertex. Then MT is positive denite.
Thm B =⇒ Thm A : start with a weakly convex, decomposable and codecomposable polyhedron P0. Choose a concave edge e1 and ll it with a simplex, to get P1 e1 is an interior edge of P1. Repeat with e2, etc. After a nite number of steps, Pn is the convex hull of P0. To go from Mk+1 to Mk , simply remove a line and a column (corresponding to ek+1). Since Mn is positive denite, so is M0. Therefore P0 is inf rigid, qed.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Concavity of S
Thm B. Let Q be a convex polyhedron, and let T be any triangulation without interior vertex. Then MT is positive denite.
Thm B =⇒ Thm A : start with a weakly convex, decomposable and codecomposable polyhedron P0. Choose a concave edge e1 and ll it with a simplex, to get P1 e1 is an interior edge of P1. Repeat with e2, etc. After a nite number of steps, Pn is the convex hull of P0. To go from Mk+1 to Mk , simply remove a line and a column (corresponding to ek+1). Since Mn is positive denite, so is M0. Therefore P0 is inf rigid, qed.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Concavity of S
Thm B. Let Q be a convex polyhedron, and let T be any triangulation without interior vertex. Then MT is positive denite.
Thm B =⇒ Thm A : start with a weakly convex, decomposable and codecomposable polyhedron P0. Choose a concave edge e1 and ll it with a simplex, to get P1 e1 is an interior edge of P1. Repeat with e2, etc. After a nite number of steps, Pn is the convex hull of P0. To go from Mk+1 to Mk , simply remove a line and a column (corresponding to ek+1). Since Mn is positive denite, so is M0. Therefore P0 is inf rigid, qed.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Concavity of S
Thm B. Let Q be a convex polyhedron, and let T be any triangulation without interior vertex. Then MT is positive denite.
Thm B =⇒ Thm A : start with a weakly convex, decomposable and codecomposable polyhedron P0. Choose a concave edge e1 and ll it with a simplex, to get P1 e1 is an interior edge of P1. Repeat with e2, etc. After a nite number of steps, Pn is the convex hull of P0. To go from Mk+1 to Mk , simply remove a line and a column (corresponding to ek+1). Since Mn is positive denite, so is M0. Therefore P0 is inf rigid, qed.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
S with interior vertices
Thm C : let P be convex, let T be a triangulation with n interior vertices. Then MT has n negative and 3n zero eigenvalues. Clearly Thm C =⇒ Thm B, so we concentrate on Thm C. Thm C is true in special cases :
if T0 has no interior vertex, and all simplices share a vertex (Izmestiev).
if T0 has exactly one interior vertex, shared by all simplices (Bobenko-Izmestiev).
To prove Thm C, show that any triangulation T can be connected to such a T0 by simple moves which respect Thm C.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
S with interior vertices
Thm C : let P be convex, let T be a triangulation with n interior vertices. Then MT has n negative and 3n zero eigenvalues. Clearly Thm C =⇒ Thm B, so we concentrate on Thm C. Thm C is true in special cases :
if T0 has no interior vertex, and all simplices share a vertex (Izmestiev).
if T0 has exactly one interior vertex, shared by all simplices (Bobenko-Izmestiev).
To prove Thm C, show that any triangulation T can be connected to such a T0 by simple moves which respect Thm C.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
S with interior vertices
Thm C : let P be convex, let T be a triangulation with n interior vertices. Then MT has n negative and 3n zero eigenvalues. Clearly Thm C =⇒ Thm B, so we concentrate on Thm C. Thm C is true in special cases :
if T0 has no interior vertex, and all simplices share a vertex (Izmestiev).
if T0 has exactly one interior vertex, shared by all simplices (Bobenko-Izmestiev).
To prove Thm C, show that any triangulation T can be connected to such a T0 by simple moves which respect Thm C.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
S with interior vertices
Thm C : let P be convex, let T be a triangulation with n interior vertices. Then MT has n negative and 3n zero eigenvalues. Clearly Thm C =⇒ Thm B, so we concentrate on Thm C. Thm C is true in special cases :
if T0 has no interior vertex, and all simplices share a vertex (Izmestiev).
if T0 has exactly one interior vertex, shared by all simplices (Bobenko-Izmestiev).
To prove Thm C, show that any triangulation T can be connected to such a T0 by simple moves which respect Thm C.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
S with interior vertices
Thm C : let P be convex, let T be a triangulation with n interior vertices. Then MT has n negative and 3n zero eigenvalues. Clearly Thm C =⇒ Thm B, so we concentrate on Thm C. Thm C is true in special cases :
if T0 has no interior vertex, and all simplices share a vertex (Izmestiev).
if T0 has exactly one interior vertex, shared by all simplices (Bobenko-Izmestiev).
To prove Thm C, show that any triangulation T can be connected to such a T0 by simple moves which respect Thm C.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Pachner moves
Def : in a 1 − 4 Pachner move, one simplex is cut into 4 simplices meeting at a vertex. Def : in a 2 − 3 Pachner move, two simplices sharing a face are replaced by 3 simplices sharing an edge. Lemma D : two triangulations T ,T ′ of P can be connected by a sequence of 1− 4 moves, 2 − 3 moves, and displacements of interior vertices. The proof is based on a deep result on another kind of moves.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Pachner moves
Def : in a 1 − 4 Pachner move, one simplex is cut into 4 simplices meeting at a vertex. Def : in a 2 − 3 Pachner move, two simplices sharing a face are replaced by 3 simplices sharing an edge. Lemma D : two triangulations T ,T ′ of P can be connected by a sequence of 1− 4 moves, 2 − 3 moves, and displacements of interior vertices. The proof is based on a deep result on another kind of moves.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Pachner moves
Def : in a 1 − 4 Pachner move, one simplex is cut into 4 simplices meeting at a vertex. Def : in a 2 − 3 Pachner move, two simplices sharing a face are replaced by 3 simplices sharing an edge. Lemma D : two triangulations T ,T ′ of P can be connected by a sequence of 1− 4 moves, 2 − 3 moves, and displacements of interior vertices. The proof is based on a deep result on another kind of moves.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Pachner moves
Def : in a 1 − 4 Pachner move, one simplex is cut into 4 simplices meeting at a vertex. Def : in a 2 − 3 Pachner move, two simplices sharing a face are replaced by 3 simplices sharing an edge. Lemma D : two triangulations T ,T ′ of P can be connected by a sequence of 1− 4 moves, 2 − 3 moves, and displacements of interior vertices. The proof is based on a deep result on another kind of moves.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Pachner moves
Def : in a 1 − 4 Pachner move, one simplex is cut into 4 simplices meeting at a vertex. Def : in a 2 − 3 Pachner move, two simplices sharing a face are replaced by 3 simplices sharing an edge. Lemma D : two triangulations T ,T ′ of P can be connected by a sequence of 1− 4 moves, 2 − 3 moves, and displacements of interior vertices. The proof is based on a deep result on another kind of moves.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Pachner moves
Def : in a 1 − 4 Pachner move, one simplex is cut into 4 simplices meeting at a vertex. Def : in a 2 − 3 Pachner move, two simplices sharing a face are replaced by 3 simplices sharing an edge. Lemma D : two triangulations T ,T ′ of P can be connected by a sequence of 1− 4 moves, 2 − 3 moves, and displacements of interior vertices. The proof is based on a deep result on another kind of moves.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Def : a simplex starring move is a 1 − 4 Pachner move. In a face starring move, a 2-face is cut into 3 triangles. In an edge starring move, an edge is cut into 2 seg- ments. Thm (Morelli 1996 ; Wlodarczyk 1997) : any T ,T ′ can be connected by a se- quence of starring moves. Motivation : birational maps between to- ric varieties. Note that the statement is geometric ra- ther than topological. A topological sta- tement is much easier. Lemma D (connectedness by Pachner moves) follows from this thm. All starring moves can be realized by Pachner moves and displacements of the interior vertices.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Def : a simplex starring move is a 1 − 4 Pachner move. In a face starring move, a 2-face is cut into 3 triangles. In an edge starring move, an edge is cut into 2 seg- ments. Thm (Morelli 1996 ; Wlodarczyk 1997) : any T ,T ′ can be connected by a se- quence of starring moves. Motivation : birational maps between to- ric varieties. Note that the statement is geometric ra- ther than topological. A topological sta- tement is much easier. Lemma D (connectedness by Pachner moves) follows from this thm. All starring moves can be realized by Pachner moves and displacements of the interior vertices.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Def : a simplex starring move is a 1 − 4 Pachner move. In a face starring move, a 2-face is cut into 3 triangles. In an edge starring move, an edge is cut into 2 seg- ments. Thm (Morelli 1996 ; Wlodarczyk 1997) : any T ,T ′ can be connected by a se- quence of starring moves. Motivation : birational maps between to- ric varieties. Note that the statement is geometric ra- ther than topological. A topological sta- tement is much easier. Lemma D (connectedness by Pachner moves) follows from this thm. All starring moves can be realized by Pachner moves and displacements of the interior vertices.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Def : a simplex starring move is a 1 − 4 Pachner move. In a face starring move, a 2-face is cut into 3 triangles. In an edge starring move, an edge is cut into 2 seg- ments. Thm (Morelli 1996 ; Wlodarczyk 1997) : any T ,T ′ can be connected by a se- quence of starring moves. Motivation : birational maps between to- ric varieties. Note that the statement is geometric ra- ther than topological. A topological sta- tement is much easier. Lemma D (connectedness by Pachner moves) follows from this thm. All starring moves can be realized by Pachner moves and displacements of the interior vertices.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Def : a simplex starring move is a 1 − 4 Pachner move. In a face starring move, a 2-face is cut into 3 triangles. In an edge starring move, an edge is cut into 2 seg- ments. Thm (Morelli 1996 ; Wlodarczyk 1997) : any T ,T ′ can be connected by a se- quence of starring moves. Motivation : birational maps between to- ric varieties. Note that the statement is geometric ra- ther than topological. A topological sta- tement is much easier. Lemma D (connectedness by Pachner moves) follows from this thm. All starring moves can be realized by Pachner moves and displacements of the interior vertices.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Def : a simplex starring move is a 1 − 4 Pachner move. In a face starring move, a 2-face is cut into 3 triangles. In an edge starring move, an edge is cut into 2 seg- ments. Thm (Morelli 1996 ; Wlodarczyk 1997) : any T ,T ′ can be connected by a se- quence of starring moves. Motivation : birational maps between to- ric varieties. Note that the statement is geometric ra- ther than topological. A topological sta- tement is much easier. Lemma D (connectedness by Pachner moves) follows from this thm. All starring moves can be realized by Pachner moves and displacements of the interior vertices.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Def : a simplex starring move is a 1 − 4 Pachner move. In a face starring move, a 2-face is cut into 3 triangles. In an edge starring move, an edge is cut into 2 seg- ments. Thm (Morelli 1996 ; Wlodarczyk 1997) : any T ,T ′ can be connected by a se- quence of starring moves. Motivation : birational maps between to- ric varieties. Note that the statement is geometric ra- ther than topological. A topological sta- tement is much easier. Lemma D (connectedness by Pachner moves) follows from this thm. All starring moves can be realized by Pachner moves and displacements of the interior vertices.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Def : a simplex starring move is a 1 − 4 Pachner move. In a face starring move, a 2-face is cut into 3 triangles. In an edge starring move, an edge is cut into 2 seg- ments. Thm (Morelli 1996 ; Wlodarczyk 1997) : any T ,T ′ can be connected by a se- quence of starring moves. Motivation : birational maps between to- ric varieties. Note that the statement is geometric ra- ther than topological. A topological sta- tement is much easier. Lemma D (connectedness by Pachner moves) follows from this thm. All starring moves can be realized by Pachner moves and displacements of the interior vertices.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Def : a simplex starring move is a 1 − 4 Pachner move. In a face starring move, a 2-face is cut into 3 triangles. In an edge starring move, an edge is cut into 2 seg- ments. Thm (Morelli 1996 ; Wlodarczyk 1997) : any T ,T ′ can be connected by a se- quence of starring moves. Motivation : birational maps between to- ric varieties. Note that the statement is geometric ra- ther than topological. A topological sta- tement is much easier. Lemma D (connectedness by Pachner moves) follows from this thm. All starring moves can be realized by Pachner moves and displacements of the interior vertices.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Def : a simplex starring move is a 1 − 4 Pachner move. In a face starring move, a 2-face is cut into 3 triangles. In an edge starring move, an edge is cut into 2 seg- ments. Thm (Morelli 1996 ; Wlodarczyk 1997) : any T ,T ′ can be connected by a se- quence of starring moves. Motivation : birational maps between to- ric varieties. Note that the statement is geometric ra- ther than topological. A topological sta- tement is much easier. Lemma D (connectedness by Pachner moves) follows from this thm. All starring moves can be realized by Pachner moves and displacements of the interior vertices.
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Proof of Thm C from Lemma D
The statement of thm C is invariant under Pachner moves and displacements of interior vertices. Thm C : let P be convex, let T be a triangulation with n interior vertices. Then MT has n negative and 3n zero eigenvalues. Displacements : by inf rigidity (Dehn) the corank is 3n. Therefore the signature does not change. For 2−3 moves : consider the change in MT in a 2−3 move. Adds one line/column. Adds a 10 × 10 matrix, (∂(θ′i − θi )/∂li ). Inf rigidity of bipyramid =⇒ this matrix has rank 1. Check that its non-zero eigenvalue is positive.
Same type of argument for 1− 4 moves (rank 2).
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Proof of Thm C from Lemma D
The statement of thm C is invariant under Pachner moves and displacements of interior vertices. Thm C : let P be convex, let T be a triangulation with n interior vertices. Then MT has n negative and 3n zero eigenvalues. Displacements : by inf rigidity (Dehn) the corank is 3n. Therefore the signature does not change. For 2−3 moves : consider the change in MT in a 2−3 move. Adds one line/column. Adds a 10 × 10 matrix, (∂(θ′i − θi )/∂li ). Inf rigidity of bipyramid =⇒ this matrix has rank 1. Check that its non-zero eigenvalue is positive.
Same type of argument for 1− 4 moves (rank 2).
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Proof of Thm C from Lemma D
The statement of thm C is invariant under Pachner moves and displacements of interior vertices. Thm C : let P be convex, let T be a triangulation with n interior vertices. Then MT has n negative and 3n zero eigenvalues. Displacements : by inf rigidity (Dehn) the corank is 3n. Therefore the signature does not change. For 2−3 moves : consider the change in MT in a 2−3 move. Adds one line/column. Adds a 10 × 10 matrix, (∂(θ′i − θi )/∂li ). Inf rigidity of bipyramid =⇒ this matrix has rank 1. Check that its non-zero eigenvalue is positive.
Same type of argument for 1− 4 moves (rank 2).
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Proof of Thm C from Lemma D
The statement of thm C is invariant under Pachner moves and displacements of interior vertices. Thm C : let P be convex, let T be a triangulation with n interior vertices. Then MT has n negative and 3n zero eigenvalues. Displacements : by inf rigidity (Dehn) the corank is 3n. Therefore the signature does not change. For 2−3 moves : consider the change in MT in a 2−3 move. Adds one line/column. Adds a 10 × 10 matrix, (∂(θ′i − θi )/∂li ). Inf rigidity of bipyramid =⇒ this matrix has rank 1. Check that its non-zero eigenvalue is positive.
Same type of argument for 1− 4 moves (rank 2).
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Proof of Thm C from Lemma D
The statement of thm C is invariant under Pachner moves and displacements of interior vertices. Thm C : let P be convex, let T be a triangulation with n interior vertices. Then MT has n negative and 3n zero eigenvalues. Displacements : by inf rigidity (Dehn) the corank is 3n. Therefore the signature does not change. For 2−3 moves : consider the change in MT in a 2−3 move. Adds one line/column. Adds a 10 × 10 matrix, (∂(θ′i − θi )/∂li ). Inf rigidity of bipyramid =⇒ this matrix has rank 1. Check that its non-zero eigenvalue is positive.
Same type of argument for 1− 4 moves (rank 2).
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Proof of Thm C from Lemma D
The statement of thm C is invariant under Pachner moves and displacements of interior vertices. Thm C : let P be convex, let T be a triangulation with n interior vertices. Then MT has n negative and 3n zero eigenvalues. Displacements : by inf rigidity (Dehn) the corank is 3n. Therefore the signature does not change. For 2−3 moves : consider the change in MT in a 2−3 move. Adds one line/column. Adds a 10 × 10 matrix, (∂(θ′i − θi )/∂li ). Inf rigidity of bipyramid =⇒ this matrix has rank 1. Check that its non-zero eigenvalue is positive.
Same type of argument for 1− 4 moves (rank 2).
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Proof of Thm C from Lemma D
The statement of thm C is invariant under Pachner moves and displacements of interior vertices. Thm C : let P be convex, let T be a triangulation with n interior vertices. Then MT has n negative and 3n zero eigenvalues. Displacements : by inf rigidity (Dehn) the corank is 3n. Therefore the signature does not change. For 2−3 moves : consider the change in MT in a 2−3 move. Adds one line/column. Adds a 10 × 10 matrix, (∂(θ′i − θi )/∂li ). Inf rigidity of bipyramid =⇒ this matrix has rank 1. Check that its non-zero eigenvalue is positive.
Same type of argument for 1− 4 moves (rank 2).
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Proof of Thm C from Lemma D
The statement of thm C is invariant under Pachner moves and displacements of interior vertices. Thm C : let P be convex, let T be a triangulation with n interior vertices. Then MT has n negative and 3n zero eigenvalues. Displacements : by inf rigidity (Dehn) the corank is 3n. Therefore the signature does not change. For 2−3 moves : consider the change in MT in a 2−3 move. Adds one line/column. Adds a 10 × 10 matrix, (∂(θ′i − θi )/∂li ). Inf rigidity of bipyramid =⇒ this matrix has rank 1. Check that its non-zero eigenvalue is positive.
Same type of argument for 1− 4 moves (rank 2).
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Proof of Thm C from Lemma D
The statement of thm C is invariant under Pachner moves and displacements of interior vertices. Thm C : let P be convex, let T be a triangulation with n interior vertices. Then MT has n negative and 3n zero eigenvalues. Displacements : by inf rigidity (Dehn) the corank is 3n. Therefore the signature does not change. For 2−3 moves : consider the change in MT in a 2−3 move. Adds one line/column. Adds a 10 × 10 matrix, (∂(θ′i − θi )/∂li ). Inf rigidity of bipyramid =⇒ this matrix has rank 1. Check that its non-zero eigenvalue is positive.
Same type of argument for 1− 4 moves (rank 2).
Jean-Marc Schlenker Problèmes anciens et nouveaux sur les polyèdres euclidiens
The beginning of the story The Renaissance
Modern times Open questions A rigidity result
Proof of Thm C from Lemma D
The statement of thm C is invariant under Pachner moves and displacements of inte