The $2$-planes $\\{\mathcal{H}_{j}\\}_{j\in\mathbb{Z}}$ are tangent to the
foregoing conic. Symmetrically, the $2$-planes
$\\{\mathcal{V}_{i}\\}_{i\in\mathbb{Z}}$ are tangent to a conic in the plane
at infinity of $\overline{\mathbb{R}^{3}}$. ∎

For smooth surfaces in $\mathbb{R}^{3}$ such that both families of curvature
lines are planar, the planes of one family of planar curvature lines are
tangent to a cylinder and the planes of the second family of curvature lines
are also tangent to a cylinder. In other words, each family of planes is
concurrent at a point in the plane at infinity of $\overline{\mathbb{R}^{3}}$.
The two cylinders have orthogonal directions. See [50, 12] for more
information. The planes $\\{\mathcal{H}_{j}\\}_{j\in\mathbb{Z}}$ (resp.
$\\{\mathcal{V}_{i}\\}_{i\in\mathbb{Z}}$) in the case $n=3$ of Proposition
4.29 are generically not concurrent at a point in the plane at infinity of
$\overline{\mathbb{R}^{3}}$. Instead, they are tangent planes of a conic that
is contained in the plane at infinity of $\overline{\mathbb{R}^{3}}$.

Recently, a discretisation of surfaces with planar curvature lines was
introduced in [53]. It is a discretisation that is based on the classical fact
that the Gauss map is a Combescure transformation of curvature-line
parametrisations. The discretisation in [53] concerns circular nets with
planar parameter lines that are more restrictive than the circular nets in
Proposition 4.28 and Proposition 4.29.

##  5 Principal Contact Element Nets with Spherical Parameter Lines

###  5.1 Lie Geometry and Principal Contact Element Nets

We start with some standard facts about the Lie geometry of oriented
hyperspheres in Euclidean space $\mathbb{R}^{n}$. See [15, 3, 9] for more
information on Lie geometry.

Let $e_{1},\ldots,e_{n+1},e_{n+2},e_{n+3}$ be an orthonormal basis in
$\mathbb{R}^{n+1,2}$:

$\displaystyle\langle e_{i},e_{j}\rangle=0,\ i\neq j,\langle
e_{i},e_{i}\rangle=1,\ i=1,\ldots,n+1,\langle e_{n+2},e_{n+2}\rangle=\langle
e_{n+3},e_{n+3}\rangle=-1.$

The norm of a vector $x=(x_{1},\ldots,x_{n+3})=\sum_{i=1}^{n+3}x_{i}e_{i}$ is
given by

$\langle x,x\rangle=\sum_{i=1}^{n+1}x_{i}^{2}-x_{n+2}^{2}-x_{n+3}^{2}.$

It is convenient to introduce two isotropic vectors

$e_{\infty}=\frac{1}{2}(e_{n+2}+e_{n+1}),\quad
e_{0}=\frac{1}{2}(e_{n+2}-e_{n+1}).$ (5)

The $(n+1)$-dimensional _Lie quadric_

$\mathcal{L}^{n+1}:=\\{[x]=[x_{1}\ldots,x_{n+3}]\in\mathbb{R}\mathrm{P}^{n+2}\mid\langle
x,x\rangle=0\\}$

parametrises the set of non-empty oriented hyperspheres, oriented hyperplanes
and points in Euclidean space $\mathbb{R}^{n}$. Models of the above elements
in $\mathcal{L}^{n+1}$ are as follows.

  * •

Oriented hypersphere with center $c\in\mathbb{R}^{n}$ and signed radius
$r\in\mathbb{R}$:

$\hat{S}=[c+e_{0}+(|c|_{\mathbb{R}^{n}}^{2}-r^{2})e_{\infty}+re_{n+3}].$ (6)

  * •

Oriented hyperplane $\langle v,x\rangle_{\mathbb{R}^{n}}=d$ with
$|v|_{\mathbb{R}^{n}}=1$ and $d\in\mathbb{R}$:

$\hat{E}=[v+0\cdot e_{0}+2de_{\infty}+e_{n+3}].$ (7)

  * •

Point $P\in\mathbb{R}^{n}$:

$\hat{P}=[P+e_{0}+|P|_{\mathbb{R}^{n}}^{2}e_{\infty}+0\cdot e_{n+3}].$ (8)

The Möbius quadric $\mathcal{M}^{n}:=\\{[x]\in\mathbb{R}\mathrm{P}^{n+1}\mid
x_{1}^{2}+\ldots+x_{n+1}^{2}-x_{n+2}^{2}=0\\}$ is identified with
$\mathcal{L}^{n+1}\cap[e_{n+3}]^{\perp}$ where $[e_{n+3}]^{\perp}$ is the
polar hyperplane relative to $\mathcal{L}^{n+1}$ of the point $[e_{n+3}]$.
Points of the _point complex_ $\mathcal{L}^{n+1}\cap[e_{n+3}]^{\perp}$ are
identified with spheres of radius zero, which are treated as points in
$\mathbb{R}^{n}$. The point $[e_{\infty}]$ is interpreted as the point at
infinity of the one-point compactification of $\mathbb{R}^{n}$. Points of the
_hyperplane complex_ $\mathcal{L}^{n+1}\cap[e_{\infty}]^{\perp}$, where
$[e_{\infty}]^{\perp}$ is the polar hyperplane relative to $\mathcal{L}^{n+1}$
of the point $[e_{\infty}]$, are identified with oriented hyperplanes in
$\mathbb{R}^{n}$.

The Lie quadric provides a double cover of the points in the exterior of
$\mathcal{M}^{n}$. In Möbius geometry, the exterior of
$\mathcal{M}^{n}\subset\mathbb{R}\mathrm{P}^{n+1}$ parametrises the set of
non-oriented hyperspheres and hyperplanes in $\mathbb{R}^{n}$. Let $S$ be a
point in the exterior of the Möbius quadric $\mathcal{M}^{n},$ which is
identified with the exterior of $\mathcal{L}^{n+1}\cap[e_{n+3}]^{\perp}$.
Then, the line $S\vee[e_{n+3}]$ intersects the Lie quadric at two real points,
which are interpreted as the two orientations of the hypersphere in
$\mathbb{R}^{n}$ corresponding to the point $S$.

Two oriented hyperspheres are in _oriented contact_ if and only if the
corresponding points $\hat{S}_{1}$ and $\hat{S}_{2}$ in $\mathcal{L}^{n+1}$
are conjugate relative to $\mathcal{L}^{n+1}$. Any two hyperspheres in
oriented contact determine a _contact element_ , i.e. their point of contact
and their common tangent hyperplane. Then, the line
$\hat{S}_{1}\vee\hat{S}_{2}$ is isotropic, i.e. lies entirely in
$\mathcal{L}^{n+1}$. An isotropic line $l$ contains exactly one point
$l\cap[e_{n+3}]^{\perp}$ with vanishing $e_{n+3}$-component (the common point
$P$ of contact of all the oriented hyperspheres in $\mathbb{R}^{n}$
corresponding to the points of $l$), and, exactly one point
$l\cap[e_{\infty}]^{\perp}$ with vanishing $e_{0}$-component (the common
tangent oriented hyperplane $E$ in $\mathbb{R}^{n}$ of all the hyperspheres).
We denote the space of isotropic lines by

$\mathcal{L}^{n+1}_{0}=\\{\text{space of isotropic lines}\\}=\\{\text{space of
contact elements}\\}.$

Principal contact element nets provide a discretisation of curvature-line
parametrisations of surfaces in $\mathbb{R}^{3}$ in terms of points and also
tangent planes. They were introduced in [8] to provide a Lie-geometric
discretisation of curvature-line parametrisations, see also [9] for more
information on principal contact element nets.

Let

$l:\mathbb{Z}^{2}\to\mathcal{L}^{4}_{0}$

be a _principal contact element net_. This means that $l$ is a _discrete line
congruence_ , i.e. any two neighboring lines intersect. By intersecting a
principal contact element net by two hyperplanes one obtains two Q-nets. The
first one

$\hat{P}:=l\cap[e_{6}]^{\perp}:\mathbb{Z}^{2}\to\mathcal{M}^{3}=\mathcal{L}^{4}\cap[e_{6}]^{\perp}$

is a Q-net in the Möbius quadric. Via stereographic projection, it determines
a circular net $P:\mathbb{Z}^{2}\to\mathbb{R}^{3}$. The second one

$\hat{E}:=l\cap[e_{\infty}]^{\perp}:\mathbb{Z}^{2}\to\mathcal{L}^{4}\cap[e_{\infty}]^{\perp}$

is a $Q$-net in the plane complex. Let $E_{i,j}$ be the oriented plane in
$\mathbb{R}^{3}$ corresponding to $\hat{E}_{i,j}$. Then,
$E:\mathbb{Z}^{2}\to\\{\text{oriented planes in }\mathbb{R}^{3}\\}$ is a
conical net. The oriented planes $E_{i,j}$, $E_{i+1,j}$, $E_{i+1,j+1}$,
$E_{i,j+1}$ are tangent to a cone of revolution. See [35, 43] for more
information concerning conical nets and their applications in architectural
geometry. Points of the circular net lie in the planes of the conical net,
i.e. $P_{i,j}\in E_{i,j}$.

In this elementary interpretation, a principal contact element net can be
identified with a pair of maps $(P,E)$ where
$P:\mathbb{Z}^{2}\to\mathbb{R}^{3}$ is a circular net and
$E:\mathbb{Z}^{2}\to\\{\text{oriented planes in }\mathbb{R}^{3}\\}$ is a
conical net such that $P_{i,j}\in E_{i,j}$. A contact element
$(P_{i,j},E_{i,j})$ has oriented contact with an oriented sphere $\mathcal{S}$
if $P_{i,j}\in\mathcal{S}$ and $E_{i,j}$ is an oriented tangent plane of
$\mathcal{S}$. The defining property of a principal contact element net
$(P,E)$ is that any two neighbouring contact elements have oriented contact
with an oriented sphere.

As explained in [9, 45] any circular net $P:\mathbb{Z}^{2}\to\mathbb{R}^{3}$
can be extended to a principal contact element net $(P,E)$ in a $2$-parameter
family of ways. There is a $2$-parameter family of possible choices for the
oriented plane $E_{0,0}$ containing $P_{0,0}$. The conical net
$E:\mathbb{Z}^{2}\to\\{\text{oriented planes in }\mathbb{R}^{3}\\}$ is
uniquely determined by the choice of $E_{0,0}$. For instance, the plane
$E_{1,0}$ is determined as the reflection of $E_{0,0}$ about the plane that is
the perpendicular bisector of the segment joining $P_{0,0}$ and $P_{1,0}$.
Similarly, any conical net $E:\mathbb{Z}^{2}\to\\{\text{oriented planes in
}\mathbb{R}^{3}\\}$ has a $2$-parameter family of extensions to a principal
contact element net $(P,E)$.

###  5.2 One Family Spherical

Let $l:\mathbb{Z}^{2}\to\mathcal{L}_{0}^{4}$ be a principal contact element
net. Let $\hat{P}:\mathbb{Z}^{2}\to\mathcal{L}^{4}\cap[e_{6}]^{\perp}$ be the
corresponding circular net in the Möbius quadric $\mathcal{M}^{3}$ which is
identified with $\mathcal{L}^{4}\cap[e_{6}]^{\perp}$. Suppose that
$\\{\hat{P}_{i,j}\\}_{i\in\mathbb{Z}}$ is a spherical parameter line, i.e.
$\vee_{i}\hat{P}_{i,j}$ is $3$-dimensional. Then, $\vee_{i}l_{i,j}$ is
$4$-dimensional because $l$ is a discrete line congruence. This motivates
Definition 5.1.

######  Definition 5.1.

Let $l:\mathbb{Z}^{2}\to\mathcal{L}_{0}^{4}$ be a principal contact element
net. The parameter line $\\{l_{i,j}\\}_{i\in\mathbb{Z}}$ is _spherical_ if
$\vee_{i}l_{i,j}$ is $4$-dimensional.

######  Proposition 5.2.

Let $l:\mathbb{Z}^{2}\to\mathcal{L}_{0}^{4}$ be a principal contact element
net. Let $P:\mathbb{Z}^{2}\to\mathbb{R}^{3}$ and
$E:\mathbb{Z}^{2}\to\\{\text{oriented planes in }\mathbb{R}^{3}\\}$ be the
corresponding circular and conical nets. Then, the following conditions are
equivalent:

  * •

$\\{l_{i,j}\\}_{i\in\mathbb{Z}}$ is spherical,

  * •

$\\{P_{i,j}\\}_{i\in\mathbb{Z}}$ is spherical, i.e. the points lie on a sphere
$\mathcal{S}_{1}$,

  * •

$\\{E_{i,j}\\}_{i\in\mathbb{Z}}$ are in oriented contact with a common
oriented sphere $\mathcal{S}_{2}$.

The planes $\\{E_{i,j}\\}_{i\in\mathbb{Z}}$ intersect the sphere
$\mathcal{S}_{1}$ at a constant angle. The spheres $\mathcal{S}_{1}$ and
$\mathcal{S}_{2}$ are concentric.

###### Proof.

Since $l_{i,j}$ is a line congruence and
$\hat{P}_{i,j}\in\mathcal{L}^{4}\cap[e_{6}]^{\perp}$,
$\hat{E}_{i,j}\in\mathcal{L}^{4}\cap[e_{\infty}]^{\perp}$, the conditions
$\dim\vee_{i}l_{i,j}=4$, $\dim\vee_{i}\hat{P}_{i,j}=3$ and
$\dim\vee_{i}\hat{E}_{i,j}=3$ are equivalent. This proves the equivalences in
Proposition 5.2.

Suppose that the points $\\{P_{i,j}\\}_{i\in\mathbb{Z}}$ are contained in a
sphere $\mathcal{S}_{1}$. The planes $E_{i,j}$ and $E_{i+1,j}$ are symmetric
about the plane that perpendicularly bisects the segment through $P_{i,j}$ and
$P_{i+1,j}$. The foregoing plane contains the centre of $\mathcal{S}_{1}$. So,
the planes $E_{i,j}$ and $E_{i+1,j}$ intersect $\mathcal{S}_{1}$ at a constant
angle. ∎

Proposition 5.2 is a discrete analogue of Joachimsthal’s theorem. The latter
states that the tangent planes of a surface in $\mathbb{R}^{3}$ along a
spherical curvature line intersect the sphere containing the spherical
curvature line at a constant angle.

######  Proposition 5.3.

Let $l:\mathbb{Z}^{2}\to\mathcal{L}_{0}^{4}$ be a principal contact element
net with spherical parameter lines $\\{l_{i,j}\\}_{i\in\mathbb{Z}}$. There
exist quadrics $\mathcal{Q}_{j}$ such that the planes $\\{l_{i,j}\vee
l_{i,j+1}\\}_{i\in\mathbb{Z}}$ are isotropic planes of $\mathcal{Q}_{j}$.
Generically, there is an alternation phenomenon: the isotropic planes
$\\{l_{i,j}\vee l_{i,j+1}\\}_{i\in\mathbb{Z}}$ decompose into the two systems
of generators $\\{l_{2i,j}\vee l_{2i,j+1}\\}_{i\in\mathbb{Z}}$ and
$\\{l_{2i+1,j}\vee l_{2i+1,j+1}\\}_{i\in\mathbb{Z}}$ of $\mathcal{Q}_{j}$.

###### Proof.

Consider the hyperplanes $\vee_{i}l_{i,j}$ and $\vee l_{i,j+1}$ in
$\mathbb{R}\mathrm{P}^{5}$. By Lemma 3.10, there is a unique pencil of
$4$-dimensional quadrics that contain the $3$-dimensional quadrics
$\vee_{i}l_{i,j}\cap\mathcal{L}^{4}$ and
$\vee_{i}l_{i,j+1}\cap\mathcal{L}^{4}$. Let $\mathcal{Q}_{j}$ be the quadric
in the pencil such that it contains a point in the plane $l_{0,j}\vee
l_{0,j+1}$ that is not contained in $l_{0,j}\cup l_{0,j+1}$. Then,
$l_{0,j}\vee l_{0,j+1}$ is an isotropic plane of $\mathcal{Q}_{j}$. Then, the
intersection $(l_{1,j}\vee l_{1,j+1})\cap\mathcal{Q}_{j}$ contains the line
$(l_{0,j}\vee l_{0,j+1})\cap(l_{1,j}\vee l_{1,j+1})$. It also contains the
lines $l_{1,j}$ and $l_{1,j+1}$. Any planar section of $\mathcal{Q}_{j}$ is
either a conic or an isotropic plane. Therefore, $l_{1,j}\vee l_{1,j+1}$ is an
isotropic plane of $\mathcal{Q}_{j}$. Iterating, we see that the planes
$l_{i,j}\vee l_{i,j+1}$ are isotropic planes of $\mathcal{Q}_{j}$.
Generically, the planes are not concurrent. So, they are isotropic planes of a
quadric of signature $(+++---)$. Since $l$ is a discrete line congruence
$(l_{i,j}\vee l_{i,j+1})\cap(l_{i+1,j}\vee l_{i+1,j+1})$ is $1$-dimensional.
By Proposition 3.7, $l_{i,j}\vee l_{i,j+1}$ and $l_{i+1,j}\vee l_{i+1,j+1}$
are in different systems of generators. ∎

The alternation phenomenon indicates that additional specifications are
required to model surfaces with spherical curvature lines, and to obtain them
in a smooth limit. This can be achieved by requiring that for any $j$ the
planes $\\{l_{i,j}\vee l_{i,j+1}\\}_{i\in\mathbb{Z}}$ are concurrent, i.e.
$\cap_{i}l_{i,j}\vee l_{i,j+1}$ is a point. This motivates the following
definition.

######  Definition 5.4.

Let $l:\mathbb{Z}^{2}\to\mathcal{L}_{0}^{4}$ be a principal contact element
net. It has _one family of spherical parameter lines_
$\\{l_{i,j}\\}_{i\in\mathbb{Z}}$ if, for any $j$ the parameter line
$\\{l_{i,j}\\}_{i\in\mathbb{Z}}$ is spherical and the planes $\\{l_{i,j}\vee
l_{i,j+1}\\}_{i\in\mathbb{Z}}$ are concurrent.

######  Proposition 5.5.

Let $l:\mathbb{Z}^{2}\to\mathcal{L}_{0}^{4}$ be a principal contact element
net with one family of spherical parameter lines
$\\{l_{i,j}\\}_{i\in\mathbb{Z}}$. Then for any $j_{0}$ and $j_{1}$, there is a
projective transformation
$f_{j_{0},j_{1}}:\vee_{i}l_{i,j_{0}}\to\vee_{i}l_{i,j_{1}}$ such that
$f_{j_{0},j_{1}}(l_{i,j_{0}})=l_{i,j_{1}}$.

###### Proof.

Let $f_{j,j+1}:\vee_{i}l_{i,j}\to\vee_{i}l_{i,j+1}$ be the central projection
with centre $\cap_{i}(l_{i,j}\vee l_{i,j+1})$. This is a projective
transformation identifying the corresponding lines:
$f_{j,j+1}(l_{i,j})=l_{i,j+1}$. The composition
$f_{j_{1}-1,j_{1}}\circ\ldots\circ f_{j_{0}+1,j_{0}+2}\circ f_{j_{0},j_{0}+1}$
is the required projective transformation $f_{j_{0},j_{1}}$. ∎

Proposition 5.5 sheds light on the classical theory of smooth surfaces with
one family of spherical curvature lines. Let
$l:\mathbb{Z}^{2}\to\\{\text{isotropic lines of }\mathcal{L}^{4}\\}$ be a
principal contact element net with one family of spherical parameter lines.
Each spherical parameter line corresponds to a $4$-dimensional space that
intersects $\mathcal{L}^{4}$ in a quadric that generically has signature
$(+++--)$. Each of these quadrics can be identified with the $3$-dimensional
Lie quadric $\mathcal{L}^{3}$ that parametrises the set of oriented circles in
the $2$-dimensional Möbius quadric $\mathcal{M}^{2}$, which can be identified
with each of the spheres in $\mathbb{R}^{3}$ that contain the spherical
parameter lines. Then, Proposition 5.5 says that any two spherical parameter
lines of a principal contact element net with one family of spherical
parameter lines are related by a Lie transformation. This observation has a
counterpart in the smooth theory. If a surface has one family of spherical
curvature lines, then any two spherical curvature lines are related by a Lie
transformation [3].

Proposition 5.6 concerns discrete line congruences with terminating Laplace
sequences. See Definition 2.4.

######  Proposition 5.6.

Let $l:\mathbb{Z}^{2}\to\mathcal{L}_{0}^{4}$ be a principal contact element
net with one family of spherical parameter lines
$\\{l_{i,j}\\}_{i\in\mathbb{Z}}$. Generically, $\mathcal{L}_{A}^{3}l$ is
Goursat degenerate and $\mathcal{L}_{B}^{2}l$ is Laplace degenerate.

###### Proof.

Let $V_{i,j}$ be the $3$-dimensional space $\vee_{i}l_{i,j}\vee l_{i,j+1}\vee
l_{i,j+2}$. The intersection $\cap_{i}V_{i,j}$ is generically $1$-dimensional.
It is the line joining the points $\cap_{i}l_{i,j}\vee l_{i,j+1}$ and
$\cap_{i}l_{i,j+1}\vee l_{i,j+2}$. The foregoing line equals
$\mathcal{L}_{B}^{2}l(i,j)$ for all $i$. So, $\mathcal{L}_{B}^{2}l$ is Laplace
degenerate. Let $H_{j}$ be the $4$-dimensional space $\vee_{i}l_{i,j}$. The
intersection $H_{j}\cap H_{j+1}\cap H_{j+2}\cap H_{j+3}$ is generically
$1$-dimensional. The foregoing line equals $\mathcal{L}_{A}^{3}l(i,j)$ for all
$i$. So, $\mathcal{L}_{A}^{3}l$ is Goursat degenerate. ∎

Let $\hat{P}:\mathbb{Z}^{2}\to\mathcal{M}^{3}$ be the circular net
corresponding to the principal contact element net $l$ with one family of
spherical parameter lines. Intersecting $\mathcal{L}^{4}$ by $[e_{6}]^{\perp}$
we obtain from Proposition 5.6 the corresponding statements for $\hat{P}$.
Thus, for the corresponding circular net $P:\mathbb{Z}^{2}\to\mathbb{R}^{3}$
we obtain that $\mathcal{L}_{A}^{3}P$ is Goursat degenerate and
$\mathcal{L}_{B}^{2}P$ is Laplace degenerate. As already mentioned in Section
4, these properties are in complete agreement with the smooth theory. We use
them to define circular nets with one family of spherical parameter lines.

Denote by $\Pi_{i,j}$ the plane containing the points
$P_{i,j},P_{i,j+1},P_{i,j+2}$.

######  Definition 5.7.

A circular net $P:\mathbb{Z}^{2}\to\mathbb{R}^{3}$ has _one family of
spherical parameter lines_ $\\{P_{i,j}\\}_{i\in\mathbb{Z}}$ if all parameter
lines $\\{P_{i,j}\\}_{i\in\mathbb{Z}}$ are spherical, and the planes
$\\{\Pi_{i,j}\\}_{i\in\mathbb{Z}}$ are concurrent.

Denote by $\mathcal{C}_{i,j}\subset\Pi_{i,j}$ the circle containing the points
$P_{i,j},P_{i,j+1},P_{i,j+2}$, and by $\mathcal{S}_{i,j}$ the sphere
containing the points
$P_{i,j},P_{i,j+1},P_{i,j+2},P_{i+1,j},P_{i+1,j+1},P_{i+1,j+2}$.

######  Proposition 5.8.

A circular net $P:\mathbb{Z}^{2}\to\mathbb{R}^{3}$ has one family of spherical
parameter lines $\\{P_{i,j}\\}_{i\in\mathbb{Z}}$ if and only if all its
parameter lines $\\{P_{i,j}\\}_{i\in\mathbb{Z}}$ are spherical and one of the
following equivalent conditions is satisfied:

  * •

the spheres $\\{\mathcal{S}_{i,j}\\}_{i\in\mathbb{Z}}$ have a (possibly
imaginary) common orthogonal sphere,

  * •

the circles $\\{\mathcal{C}_{i,j}\\}_{i\in\mathbb{Z}}$ have a (possibly
imaginary) common orthogonal sphere,

  * •

the Laplace transform $\mathcal{L}_{B}^{2}\hat{P}$ is Laplace degenerate.

###### Proof.

Let $P_{j}=\cap_{i}\Pi_{i,j}$ be the intersection point of the planes
$\\{\Pi_{i,j}\\}_{i\in\mathbb{Z}}$. Consider the (possibly imaginary) sphere
$\mathcal{S}_{j}$ with the center $P_{j}$ and orthogonal to the sphere
$\mathcal{S}_{1,j}$. Since $\mathcal{S}_{j}$ is orthogonal to all planes
$\\{\Pi_{i,j}\\}_{i\in\mathbb{Z}}$ and $\mathcal{S}_{i,j},\mathcal{S}_{i+1,j}$
and $\Pi_{i+1,j}$ belong to a pencil, $\mathcal{S}_{j}$ is orthogonal to all
the spheres $\\{\mathcal{S}_{i,j}\\}_{i\in\mathbb{Z}}$. Equivalently,
$\mathcal{S}_{j}$ is orthogonal to all the circles
$\\{\mathcal{C}_{i,j}=\mathcal{S}_{i,j}\cap\Pi_{i,j}\\}_{i\in\mathbb{Z}}$.

Lift $P:\mathbb{Z}^{2}\to\mathbb{R}^{3}$ to the Möbius quadric. Then,
$\hat{P}:\mathbb{Z}^{2}\to\mathcal{M}^{3}$ is a Q-net such that each $P_{i,j}$
is the stereographic projection of $\hat{P}_{i,j}$. By Proposition 2.3,
$\mathcal{L}_{A}^{2}\hat{P}$ is Laplace degenerate if and only if the planes
$\\{\hat{P}_{i,j}\vee\hat{P}_{i,j+1}\vee\hat{P}_{i,j+2}\\}_{i\in\mathbb{Z}}$
are concurrent. Equivalently, the circles
$\\{\mathcal{C}_{i,j}\\}_{i\in\mathbb{Z}}$ have a common orthogonal sphere
which corresponds to the intersection of $\mathcal{M}^{3}$ with the polar
hyperplane of the foregoing concurrency point.

∎

The next proposition reveals some of the geometric properties of principal
contact element nets with one family of spherical parameter lines.

######  Proposition 5.9.

Let $l:\mathbb{Z}^{2}\to\mathcal{L}_{0}^{4}$ be a principal contact element
net with one family of spherical parameter lines
$\\{l_{i,j}\\}_{i\in\mathbb{Z}}$. Let $P:\mathbb{Z}^{2}\to\mathbb{R}^{3}$ and
$E:\mathbb{Z}^{2}\to\\{\text{oriented planes in }\mathbb{R}^{3}\\}$ be the
corresponding circular and conical nets, and $n_{i,j}$ be the normal line of
$E_{i,j}$ through $P_{i,j}$. Then:

  * •

the circular net $P$ has one family of spherical parameter lines in the sense
of Definition 5.7.

  * •

for each $j$ the planes $\\{n_{i,j}\vee n_{i,j+1}\\}_{i\in\mathbb{Z}}$ are
concurrent and the concurrency point is contained in the line joining the
centres of the spheres through the points $\\{P_{i,j}\\}_{i\in\mathbb{Z}}$ and
$\\{P_{i,j+1}\\}_{i\in\mathbb{Z}}$,

  * •

for each $j$ the intersection points $\\{K_{i,j}:=E_{i,j}\cap E_{i,j+1}\cap
E_{i,j+2}\\}_{i\in\mathbb{Z}}$ are coplanar.

###### Proof.

By Proposition 5.6, $\mathcal{L}_{A}^{2}l$ is Laplace degenerate. So,
$\mathcal{L}_{A}^{2}\hat{P}$ is Laplace degenerate. Then, the first claim
follows from Proposition 5.8.

Let $\pi$ be the central projection with centre $[e_{6}]$ onto the hyperplane
$[e_{6}]^{\perp}$. Recall, each point $\hat{S}\in\mathcal{L}^{4}$ corresponds
to an oriented sphere in $\mathbb{R}^{3}$. The projection $\pi(\hat{S})$ is a
point outside $\mathcal{M}^{3}=\mathcal{L}^{4}\cap[e_{6}]^{\perp}$ that
corresponds in Möbius geometry to the same sphere in $\mathbb{R}^{3}$, without
orientation. By Lemma A.1, $\sigma\circ\pi(\hat{S})$ is the centre of the
oriented sphere in $\mathbb{R}^{3}$ that corresponds to the point $\hat{S}$.
Then, $n_{i,j}=\sigma\circ\pi(l_{i,j})$ because $n_{i,j}$ is the locus of the
centres of oriented spheres of the contact element corresponding to $l_{i,j}$.
Let $O_{j}$ be the concurrency point $\cap_{i\in\mathbb{Z}}l_{i,j}\vee
l_{i,j+j}$. Then, $\sigma\circ\pi(O_{j})$ is the concurrency point
$\cap_{i\in\mathbb{Z}}n_{i,j}\vee n_{i,j+1}$.

Each space $H_{j}:=\mathrm{join}\\{l_{i,j}\\}_{i\in\mathbb{Z}}$ is
$4$-dimensional. The point $\pi(H_{j}^{\perp})$ has a polar hyperplane
relative to $\mathcal{L}^{4}$ that intersects $[e_{6}]^{\perp}$ in a
$3$-dimensional space that equals $H_{j}\cap[e_{6}]^{\perp}$. So,
$\pi(H_{j}^{\perp})$ is a point outside
$\mathcal{M}^{3}=\mathcal{L}^{4}\cap[e_{6}]^{\perp}$ that corresponds in
Möbius geometry to the sphere $\mathcal{H}_{j}$ in $\mathbb{R}^{3}$ through
the points $\\{P_{i,j}\\}_{i\in\mathbb{Z}}$. By Lemma A.1,
$\sigma\circ\pi(H_{j}^{\perp})$ is the centre of the sphere $\mathcal{H}_{j}$.
The point $O_{j}$ is conjugate to the points $\\{l_{i,j}\cap
l_{i,j+1}\\}_{i\in\mathbb{Z}}$ which span the $3$-dimensional space $H_{j}\cap
H_{j+1}$. So, the point $O_{j}$ is contained in the line $H_{j}^{\perp}\vee
H_{j+1}^{\perp}$. The concurrency point $\sigma\circ\pi(O_{j})$ is contained
in the line $\sigma\circ\pi(H_{j}^{\perp}\vee H_{j+1}^{\perp})$ which is the
line joining the centres of the spheres $\mathcal{H}_{j}$ and
$\mathcal{H}_{j+1}$.

Let $\hat{E}_{i,j}\in\mathcal{L}^{4}\cap[e_{\infty}]^{\perp}$ and
$\hat{K}_{i,j}\in\mathcal{L}^{4}\cap[e_{6}]^{\perp}$ be the lifts of the
planes $E_{i,j}$ and of their intersection points $K_{i,j}$ to the Lie
quadric. Since
$\hat{E}_{i,j},\hat{E}_{i,j+1},\hat{E}_{i,j+2}\in\hat{K}_{i,j}^{\perp}$ and
the Laplace transform $\mathcal{L}_{B}^{2}\hat{E}$ is independent of $i$, we
obtain the conjugacies $\hat{K}_{i,j}\perp\mathcal{L}_{B}^{2}\hat{E}(j)$ and
$\mathcal{L}_{B}^{2}\hat{E}(j)\perp[e_{\infty}]$ relative to
$\mathcal{L}^{4}$. Thus, the line
$\mathcal{L}_{B}^{2}\hat{E}(j)\vee[e_{\infty}]$ is contained in the plane
complex $[e_{\infty}]^{\perp}$ and intersects $\mathcal{L}^{4}$ in the point
$[e_{\infty}]$ and a second real point which is conjugate to the points
$\\{\hat{K}_{i,j}\\}_{i\in\mathbb{Z}}$. The second intersection point
corresponds to a plane in $\mathbb{R}^{3}$ that contains the points
$\\{K_{i,j}\\}_{i\in\mathbb{Z}}$.

∎

###  5.3 Both Families Spherical

######  Definition 5.10.

Let $l:\mathbb{Z}^{2}\to\mathcal{L}_{0}^{4}$ be a principal contact element
net. It has _two families of spherical parameter lines_ if the following
conditions are satisfied:

  * •

for any $j$ the parameter line $\\{l_{i,j}\\}_{i\in\mathbb{Z}}$ is spherical
and $\cap_{i}l_{i,j}\vee l_{i,j+1}$ is a point.

  * •

for any $i$ the parameter line $\\{l_{i,j}\\}_{j\in\mathbb{Z}}$ is spherical
and $\cap_{j}l_{i,j}\vee l_{i+1,j}$ is a point.

We start with Proposition 5.11, which is an analogue of Theorem 1.1 for line
congruences. It is used later in the proof of Theorem 5.12.

######  Proposition 5.11.

Let $l:[m]\times[m]\to\\{\text{lines in }\mathbb{R}\mathrm{P}^{n}\\}$ be a
discrete line congruence, where $m\geq 2$ and $n\geq 3$. Suppose that
$\mathcal{L}_{A}^{d}l:[m-d]\times[m-d]\to\mathbb{R}\mathrm{P}^{n}$ and
$\mathcal{L}_{B}^{d}l:[m-d]\times[m-d]\to\mathbb{R}\mathrm{P}^{n}$ are well
defined discrete line congruences for all $d\leq m-1$. Then,
$\mathcal{L}_{A}^{m-1}l$ and $\mathcal{L}_{B}^{m-1}l$ are two lines. Suppose
that the lines $\\{l_{i,j}\mid(i,j)\in[m]\times[m]\text{ s.t.\
}(i,j)\neq(m,m)\\}$ are contained in a quadric
$\mathcal{Q}\subset\mathbb{R}\mathrm{P}^{n}$. Then, the line $l_{m,m}$ is
contained in $\mathcal{Q}$ if and only if $\mathcal{L}_{A}^{m-1}l$ and
$\mathcal{L}_{B}^{m-1}l$ are conjugate with respect to $\mathcal{Q}$.

###### Proof.

Let $\Pi$ be a hyperplane of $\mathbb{R}\mathrm{P}^{n}$ that does not contain
any of the lines of $l$. The points $P_{i,j}:=l_{i,j}\cap\Pi$ build a Q-net
$P$ in the quadric $\mathcal{Q}\cap\Pi$. By Theorem 1.1, the point $P_{m,m}$
is contained in $\mathcal{Q}\cap\Pi$ if and only if the points
$\mathcal{L}^{m-1}_{A}P$ and $\mathcal{L}_{B}^{m-1}P$ are conjugate relative
to $\mathcal{Q}$. As $\Pi$ is arbitrary, the line $l_{m,m}$ is contained in
$\mathcal{Q}$ if and only if $\mathcal{L}^{m-1}_{A}l$ and
$\mathcal{L}_{B}^{m-1}l$ are conjugate relative to $\mathcal{Q}$. ∎

######  Theorem 5.12.

Let $l:\mathbb{Z}^{2}\to\mathcal{L}_{0}^{4}$ be a principal contact element
net with two families of spherical parameter lines. Then,
$\mathcal{L}^{2}_{A}l$ and $\mathcal{L}^{2}_{B}l$ are Laplace degenerate. Each
set of points $(\vee_{i}l_{i,j})^{\perp}$ and $(\vee_{j}l_{i,j})^{\perp}$ is
contained in a $2$-plane. The two $2$-planes are conjugate relative to
$\mathcal{L}^{4}$. Let $P:\mathbb{Z}^{2}\to\mathbb{R}^{3}$ be the
corresponding circular net. Let $\mathcal{H}_{j}$ and $\mathcal{V}_{i}$ be the
spheres through the points $\\{P_{i,j}\mid i\in\mathbb{Z}\\}$ and
$\\{P_{i,j}\mid j\in\mathbb{Z}\\}$, respectively. Then,

  * •

Each family of spheres has two common (possibly imaginary) points.

  * •

The centres of the spheres $\mathcal{H}_{j}$ are contained in a plane that is
orthogonal to the plane containing the centres of the spheres
$\mathcal{V}_{i}$.

###### Proof.

Proposition 5.6 ensures that $\mathcal{L}^{2}_{A}l$ and $\mathcal{L}^{2}_{B}l$
are Laplace degenerate. The lines
$\\{\mathcal{L}_{A}^{2}l(i)\\}_{i\in\mathbb{Z}}$ span the space
$X:=\cap_{j\in\mathbb{Z}}H_{j}$, where $H_{j}:=\mathrm{join}\\{l_{i,j}\mid
i\in\mathbb{Z}\\}$. The space $X$ is $2$-dimensional. Indeed, it is the join
of the concurrency points $\cap_{j}l_{i,j}\vee l_{i+1,j}$,
$\cap_{j}l_{i+1,j}\vee l_{i+2,j}$ and $\cap_{j}l_{i+2,j}\vee l_{i+3,j}$. The
lines $\\{\mathcal{L}_{B}^{2}l(j)\\}_{j\in\mathbb{Z}}$ span the
$2$-dimensional space $Y:=\cap_{i\in\mathbb{Z}}V_{i}$ where
$V_{i}:=\mathrm{join}\\{l_{i,j}\mid j\in\mathbb{Z}\\}$. By Proposition 5.11,
$\mathcal{L}_{A}^{2}l(i)\perp\mathcal{L}_{B}^{2}l(j)$ for any $i,j$. So, $X$
and $Y$ are conjugate planes. The points
$\\{H_{j}^{\perp}\\}_{j\in\mathbb{Z}}$ are contained in $Y$. The points
$\\{V_{i}^{\perp}\\}_{i\in\mathbb{Z}}$ are contained in $X$.

For each $j$, $H_{j}\cap\mathcal{L}^{4}\cap[e_{6}]^{\perp}$ is the sphere in
$\mathcal{M}^{3}=\mathcal{L}^{4}\cap[e_{6}]^{\perp}$ that contains the points
$\\{\hat{P}_{i,j}=l_{i,j}\cap[e_{6}]^{\perp}\\}_{i\in\mathbb{Z}}$. The
$2$-dimensional space $X$ intersects $[e_{6}]^{\perp}$ in a line that
intersects $\mathcal{M}^{3}$ in two (possibly imaginary) points that are
common to all the spheres of one family. Similarly, the second family of
spheres also has two (possibly imaginary) common points. Thus, the centres of
the spheres $\\{\mathcal{H}_{j}\\}_{j\in\mathbb{Z}}$ are contained in a plane
and the centres of the spheres $\\{\mathcal{V}_{i}\\}_{i\in\mathbb{Z}}$ are
contained in a plane. The two planes are orthogonal. Indeed, this is a
limiting case of the two confocal quadrics in Corollary 4.25. ∎

Theorem 5.12 has a direct analogue in the classical theory of smooth surfaces
with two families of spherical curvature liness. If a smooth surface in
$\mathbb{R}^{3}$ has two families of spherical curvature lines, then the
spheres correspond to points in the exterior of $\mathcal{L}^{4}$ that are
contained in two planes that are conjugate relative to $\mathcal{L}^{4}$ [3,
16].

## Acknowledgments

This research was supported by the DFG Collaborative Research Center TRR 109
“Discretization in Geometry and Dynamics” and the Berlin Mathematical School.
Thanks to the Leverhulme Trust for the grant SAS-2018-040 that enabled A. Y.
F. to initiate the work in this article. We thank Niklas Affolter and Yuri
Suris for stimulating discussions on iterated Laplace transformations. Thanks
to Théo Tyburn and Oliver Gross for making Figure 1.

##  Appendix A Möbius Geometry

Here we give a brief overview of Möbius geometry and Darboux cyclides. For
instance, see [3, 27, 9] for more information on Möbius geometry and see [18]
for more information on Darboux cyclides.

Möbius geometry is a subgeometry of Lie geometry, with points distinguishable
among all hyperspheres as those of radius zero. As it was explained in Section
5.1, it is a restriction of Lie geometry to the polar hyperplane
$[e_{n+3}]^{\perp}$ of a time-like vector $e_{n+3}$. Thus one can use the Lie-
geometric description and just omit the $e_{n+3}$-component, passing to the
Lorentz space $\mathbb{R}^{n+1,1}\subset\mathbb{R}^{n+1,2}$. The resulting
objects are points of the $(n+1)$-dimensional projective space
$\mathbb{R}\mathrm{P}^{n+1}:=\mathrm{P}(\mathbb{R}^{n+1,1})$. We continue to
use notations of Section 5.1 in the context of Möbius geometry.

The $n$-dimensional _Möbius quadric_ is given by

$\mathcal{M}^{n}:=\\{[x]=[x_{1}\ldots,x_{n+2}]\in\mathbb{R}\mathrm{P}^{n+1}\mid\langle
x,x\rangle=0\\},$

where

$\langle x,y\rangle=\sum_{i=1}^{n+1}x_{i}y_{i}-x_{n+2}y_{n+2}.$

is the Lorentz product of two vectors $x,y\in\mathbb{R}^{n+1,1}$. We will use
the basis $e_{1},\ldots,e_{n},e_{o},e_{\infty}$ (see (5)), and denote the
corresponding coordinates of the vectors $x\in\mathbb{R}^{n+1,1}$ as follows:

$x=\sum_{i=1}^{n}x_{i}e_{i}+x_{0}e_{0}+x_{\infty}e_{\infty},\quad
x_{0}=x_{n+2}-x_{n+1},\ x_{\infty}=x_{n+2}+x_{n+1}.$

Without loss of generality two arbitrary points on the Möbius quadric can be
normalized as $[e_{\infty}]$ and $[e_{0}]$. Let us consider the tangent
hyperplanes to $\mathcal{M}^{n}$ at these points as well as their intersection
plane:

$\displaystyle\mathcal{H}^{\infty}$ $\displaystyle:=$
$\displaystyle\\{[x]\in\mathbb{R}\mathrm{P}^{n+1}\mid x_{0}=0\\},$
$\displaystyle\mathcal{H}^{0}$ $\displaystyle:=$
$\displaystyle\\{[x]\in\mathbb{R}\mathrm{P}^{n+1}\mid x_{\infty}=0\\},$
$\displaystyle\mathcal{H}$ $\displaystyle:=$
$\displaystyle\mathcal{H}^{\infty}\cap\mathcal{H}^{0}=\\{[x]\in\mathbb{R}\mathrm{P}^{n+1}\mid
x_{0}=x_{\infty}=0\\}.$

Elements of Möbius geometry of $\mathbb{R}^{n}\cup\infty$ are modelled in the
space $\mathbb{R}^{n+1,1}$ of homogeneous coordinates as follows:

  * •

Point $P\in\mathbb{R}^{n}$:

$P_{\rm Euc}=P+e_{0}+|P|^{2}e_{\infty}.$ (9)

  * •

Infinity $\infty$:

$e_{\infty}.$ (10)

  * •

Hypersphere $\mathcal{S}$ with center $c\in\mathbb{R}^{n}$ and radius $r>0$:

$S_{\rm Euc}=c+e_{0}+(|c|^{2}-r^{2})e_{\infty}.$ (11)

  * •

Hyperplane $\langle v,P\rangle_{\mathbb{R}^{n}}=d$ with
$|v|_{\mathbb{R}^{n}}=1$ and $d\in\mathbb{R}$:

$E_{\rm Euc}=v+0\cdot e_{0}+2de_{\infty}.$ (12)

In the projective space $\mathbb{R}\mathrm{P}^{n+1}$ points of
$\mathbb{R}^{n}\cup\infty$ are represented by points of the Möbius quadric.
Similarly to Section 5.1 we denote the elements of the projective space
corresponding to the representatives (9, 11, 12) by:

$\hat{P}=[P_{\rm Euc}],\hat{S}=[S_{\rm Euc}],\hat{E}=[E_{\rm
Euc}]\in\mathbb{R}\mathrm{P}^{n+1}.$

Hyperspheres and hyperplanes are intersections of $\mathcal{M}^{n}$ with
hyperplanes $H$. These are represented by their space-like polar points
$\hat{S},\hat{E}\in\mathbb{R}\mathrm{P}^{n+1}$. Note, $\langle S_{\rm
Euc},S_{\rm Euc}\rangle=r^{2}$. If the hyperplane $H$ does not intersect
$\mathcal{M}^{n}$ then the corresponding hypersphere is called _imaginary_ ,
and is represented by its polar time-like vector, i.e. purely imaginary $r$.
The incidence condition $P\in S$ reads as the polarity condition $\langle
S_{\rm Euc},P_{\rm Euc}\rangle=0$. The hyperplanes are treated as spheres
passing through $\infty$, i.e. satisfying the relation $\langle E_{\rm
Euc},e_{\infty}\rangle=0$.

To sum up: Points in the exterior of $\mathcal{M}^{n}$ (where $\langle
x,x\rangle>0$) correspond either to Euclidean hyperspheres with real centres
and real radii or else to hyperplanes in $\mathbb{R}^{n}$. Hyperplanes are
interpreted as Euclidean hyperspheres passing through the infinity point.
Points in $\mathcal{M}^{n}$ (where $\langle x,x\rangle=0$) correspond to
Euclidean hyperspheres with real centres and null radii, i.e. to points of
$\mathbb{R}\cup\infty$. Points in the interior of $\mathcal{M}^{n}$ (where
$\langle x,x\rangle<0$) correspond to imaginary hyperspheres with real centres
and purely imaginary radii.

Two hyperspheres $\mathcal{S}^{\prime}$ and $\mathcal{S}^{\prime\prime}$ are
orthogonal in $\mathbb{R}^{n}$ if and only if their Möbius representatives are
orthogonal $\langle S^{\prime}_{\rm Euc},S^{\prime\prime}_{\rm Euc}\rangle=0$.
Two (possibly imaginary) hyperspheres $\mathcal{M}^{n}\cap G_{1}$ and
$\mathcal{M}^{n}\cap G_{2}$ are orthogonal if their poles $G_{1}^{\perp}$ and
$G_{2}^{\perp}$ are conjugate relative to $\mathcal{M}^{n}$. Similarly, the
intersection of $\mathcal{M}^{n}$ with an $m$-dimensional projective subspace
$H\subset\mathbb{R}\mathrm{P}^{n+1}$ is an $(m-1)$-dimensional sphere, which
is imaginary if $H\cap\mathcal{M}^{n}=\emptyset$.

Let $\sigma:\mathbb{R}\mathrm{P}^{n+1}\setminus N\to\mathcal{H}^{0}$ be the
central projection

$[x=\sum_{i}^{n}x_{i}e_{i}+x_{0}e_{0}+x_{\infty}e_{\infty}]\xrightarrow{\sigma}[\sum_{i}^{n}x_{i}e_{i}+x_{0}e_{0}]$

with center $N:=[e_{\infty}]$. The hyperplane $\mathcal{H}^{0}$ is the
projective closure of $\mathcal{H}^{0}\setminus\mathcal{H}$. The later can be
identified with $\mathbb{R}^{n}:=\\{[x]\in\mathbb{R}\mathrm{P}^{n+1}\mid
x_{0}=1,x_{\infty}=0\\}$, where each point $[P+e_{0}]$ is identified with the
point $P=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}$.

So we also will use short notations
$\overline{\mathbb{R}^{n}}:=\mathcal{H}^{0},\
\mathbb{R}^{n}:=\mathcal{H}^{0}\setminus\mathcal{H},$ and interpret
$\mathcal{H}^{0}\setminus\mathcal{H}$ as the set of points in
$\mathbb{R}^{n}$, and $\mathcal{H}^{0}$ as its projective closure. Note that
we have already identified $\mathcal{H}^{\infty}$ with the space of
hyperplanes in $\mathbb{R}^{n}$.

By comparing with (9), we see that the restriction to the Möbius quadric
$\sigma:\mathcal{M}^{n}\setminus
N\to\mathcal{H}^{0}\setminus\mathcal{H}\cong\mathbb{R}^{n}$ is given by

$\sigma(\hat{P})=[P+e_{0}]\cong P\in\mathbb{R}^{n}.$

The quadric
$\mathcal{M}_{\mathbb{C}}^{n}:=\\{[x]\in\mathbb{C}\mathrm{P}^{n+1}\mid\langle
x,x\rangle=0\\}$ is the _complexification_ of $\mathcal{M}^{n}$. The
intersection of $\mathcal{M}_{\mathbb{C}}^{n}$ with an $m$-dimensional
projective subspace in $\mathbb{C}\mathrm{P}^{n+1}$ is an $(m-1)$-dimensional
_complex sphere_ in $\mathcal{M}_{\mathbb{C}}^{n}$. Similarly to the real case
we denote by $\sigma_{\mathbb{C}}$ the central projection with center
$N_{\mathbb{C}}:=[e_{\infty}]\in\mathbb{C}\mathrm{P}^{n+1}$:

$\sigma_{\mathbb{C}}:\mathbb{C}\mathrm{P}^{n+1}\setminus
N_{\mathbb{C}}\to\mathcal{H}^{0}_{\mathbb{C}},$

and identify the points $[P+e_{0}]\cong P\in\mathbb{C}^{n}$ and the spaces
$\mathcal{H}^{0}_{\mathbb{C}}\cong\overline{\mathbb{C}^{n}},\mathcal{H}^{0}_{\mathbb{C}}\setminus\mathcal{H}_{\mathbb{C}}\cong\mathbb{C}^{n}$.

From representations (9, 11, 12) we easily obtain the following result.

######  Lemma A.1.

Let $\hat{S}\in\mathbb{C}\mathrm{P}^{n+1}$ be a point not contained in
$\mathcal{H}^{\infty}_{\mathbb{C}}$, i.e. with non-vanishing
$e_{0}$-component, and $\hat{S}^{\perp}$ its polar hyperplane. Then the image
$\sigma_{\mathbb{C}}(\hat{S}^{\perp}\cap\mathcal{M}^{n}_{\mathbb{C}})$ is the
round sphere $\sum_{i}^{n}(P_{i}-c_{i})^{2}=r^{2}$, with center $c$ and radius
$r$ given by (11). In particular $\sigma(\hat{S})$ is the center of the
sphere. Let $\hat{E}\in\mathbb{C}\mathrm{P}^{n+1}$ be a point in
$\mathcal{H}^{\infty}_{\mathbb{C}}$, i.e. with vanishing $e_{0}$-component,
and $\hat{E}^{\perp}$ its polar hyperplane. Then the image
$\sigma_{\mathbb{C}}(\hat{E}^{\perp}\cap\mathcal{M}^{n}_{\mathbb{C}})$ is the
hyperplane $\langle v,P\rangle=d$, with parameters given by (12). In
particular $\sigma(\hat{E})$ is the point at infinity of
$\overline{\mathbb{C}^{n}}$ in the direction of the normals of the hyperplane.

######  Definition A.2.

The intersection of $\mathcal{M}^{n}\subset\mathbb{R}\mathrm{P}^{n+1}$, with a
real quadric hypersurface $\mathcal{Q}\neq\mathcal{M}^{n}$ is an
$(n-1)$-dimensional _Darboux cyclide_ in $\mathcal{M}^{n}$. The central
projection $\sigma(\mathcal{Q}\cap\mathcal{M}^{n})$ is a Darboux cyclide in
$\mathbb{R}^{n}=\mathcal{H}^{0}\setminus\mathcal{H}$.

Similarly, the intersection of $\mathcal{M}_{\mathbb{C}}^{n}$ with a complex
quadric hypersurface $\mathcal{Q}\neq\mathcal{M}_{\mathbb{C}}^{n}$ is a
_complex Darboux cyclide_. The central projection
$\sigma_{\mathbb{C}}(\mathcal{Q}\cap\mathcal{M}^{n}_{\mathbb{C}})$ is a
complex Darboux cyclide in $\overline{\mathbb{C}^{n}}$.

Definition A.2 is a direct generalisation of the definition of Darboux
cyclides in [18]. $1$-dimensional and $2$-dimensional Darboux cyclides are
classical curves and surfaces that were notably studied by John Casey [13,
14], Gaston Darboux [19] and Théodore Moutard [37, 38]. $1$-dimensional
Darboux cyclides are also known as _cyclics_. Recently, $2$-dimensional
Darboux cyclides have garnered renewed interest [44, 36, 51].

Because $1$-dimensional Darboux cyclides in $\overline{\mathbb{R}^{2}}$ are
generically quartic curves with two imaginary singular points that are the
circular points at infinity, they are also known as _bicircular quartics_
[13]. A direct generalisation is that Darboux cyclides in
$\overline{\mathbb{R}^{n}}$ are generically _bihyperspherical quartics_. By
definition, an algebraic hypersurface $\mathcal{S}$ in
$\overline{\mathbb{R}^{n}}$ is _bihyperspherical_ if its complexification
$\mathcal{S}_{\mathbb{C}}$ has a singular locus that contains the
$(n-2)$-dimensional quadric

$\mathscr{C}^{n-2}:=\mathcal{M}_{\mathbb{C}}\cap\mathcal{H}_{\mathbb{C}}=\\{[x]\in\overline{\mathbb{C}^{n}}\mid
x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}=0,x_{0}=x_{\infty}=0\\}$

that is contained in the hyperplane at infinity of
$\overline{\mathbb{C}^{n}}$. For example, any pair of hyperspheres is a
bihyperspherical quartic. Indeed, hyperspheres in $\overline{\mathbb{R}^{n}}$
can be characterised as quadrics such that their complexifications are
quadrics in $\overline{\mathbb{C}^{n}}$ that contain $\mathscr{C}^{n-2}$.

Complex $2$-dimensional Darboux cyclides in $\overline{\mathbb{C}^{3}}$ are
generically quartic surfaces that contain $\mathscr{C}^{1}$ as a double conic.
More generally, quartic surfaces with a double conic were notably studied by
Ernst Kummer [31]. Quartic surfaces with a double conic are surfaces that can
be studied as the projections of the intersection surfaces of pairs of quadric
hypersurfaces in $\mathbb{C}\mathrm{P}^{4}$ [49].

######  Definition A.3.

A $1$-parameter family of quadric hypersurfaces in $\overline{\mathbb{C}^{n}}$
is a family of _confocal quadrics_ if they are the projective duals of a
pencil of quadrics such that $\mathscr{C}^{n-2}$ is the dual of a degenerate
quadric that belongs to the pencil.

Confocal quadrics in $\mathbb{R}^{n}$ are quadrics such that their
complexifications are confocal quadrics in $\mathbb{C}^{n}$.

Definition A.3 is due to Darboux, who showed its equivalence to standard
definitions of confocal quadrics [19].

Let $\mathcal{Q}\cap\mathcal{M}_{\mathbb{C}}^{n}$ be a complex Darboux
cyclide. The two distinct quadrics $\mathcal{Q}$ and
$\mathcal{M}_{\mathbb{C}}^{n}$ span a pencil of quadrics in
$\mathbb{C}\mathrm{P}^{n+1}$. By definition, the complex Darboux cyclide is
_general_ if the pencil of quadrics contains $n+2$ degenerate quadrics. This
is the maximal number of degenerate quadrics that the pencil can contain.

Theorem A.4 is a direct generalisation of properties of $1$-dimensional and
$2$-dimensional Darboux cyclides that can be found in [19, 18].

######  Theorem A.4.

Let $\mathcal{D}:=\mathcal{Q}\cap\mathcal{M}_{\mathbb{C}}^{n}$ be a general
complex Darboux cyclide. There are $n+2$ different ways to generate the
Darboux cyclide $\mathcal{D}$ as the envelope of a $(n-1)$-dimensional family
of hyperspheres in $\mathcal{M}_{\mathbb{C}}^{n}$ that are twice tangent to
$\mathcal{D}$. For each generation $i$, the hyperspheres have a common
orthogonal hypersphere $\mathcal{S}_{i}$. The $n+2$ common orthogonal
hyperspheres $\mathcal{S}_{i}$ are pairwise orthogonal. Consider the Darboux
cyclide $\sigma_{\mathbb{C}}(\mathcal{D})$ in $\overline{\mathbb{C}^{n}}$. For
each generation of $\sigma_{\mathbb{C}}(\mathcal{D})$, the centres of the
generating spheres are contained in a quadric hypersurface of
$\overline{\mathbb{C}^{n}}$. These $n+2$ quadric hypersurfaces are confocal.

###### Proof.

There are $n+2$ degenerate quadric hypersurfaces in the pencil spanned by
$\mathcal{Q}$ and $\mathcal{M}^{n}_{\mathbb{C}}$. Let
$\mathcal{K}_{1},\ldots,\mathcal{K}_{n+2}$ be the degenerate quadrics. Let
$A_{i}$ be the point that is the apex of $\mathcal{K}_{i}$. Any tangent
hyperplane of $\mathcal{K}_{i}$ intersects $\mathcal{M}^{n}_{\mathbb{C}}$ in a
hypersphere of $\mathcal{M}_{\mathbb{C}}^{n}$. The tangent hyperplane is
tangent to $\mathcal{K}_{i}$ along the points of an isotropic line that passes
through the apex $A_{i}$. The isotropic line intersects
$\mathcal{M}_{\mathbb{C}}^{n}$ in two points. The hypersphere is tangent to
$\mathcal{D}$ at the two foregoing points. The cone $\mathcal{K}_{i}$ has a
$(n-1)$-dimensional family of tangent hyperplanes. They intersect
$\mathcal{M}^{n}_{\mathbb{C}}$ in a $(n-1)$-dimensional family of hyperspheres
that are orthogonal to the hypersphere
$A_{i}^{\perp}\cap\mathcal{M}^{n}_{\mathbb{C}}$, where $A_{i}^{\perp}$ is the
polar hyperplane of $A_{i}$ relative to $\mathcal{M}^{n}_{\mathbb{C}}$.

Consider an arbitrary plane that contains two of the points
$A_{i},\ldots,A_{n+2}$. Generically, the plane intersects
$\mathcal{D}:=\mathcal{Q}\cap\mathcal{M}_{\mathbb{C}}^{n}$ in four points. The
four points determine a planar quad such that its two Laplace points are two
of the points $A_{i}$. By Lemma 3.2, the two Laplace points are conjugate
relative to $\mathcal{M}^{n}_{\mathbb{C}}$. Therefore, any two of the points
in $A_{i}$ must be conjugate relative to $\mathcal{M}^{n}_{\mathbb{C}}$.
Equivalently, the spheres
$\mathcal{S}_{i}=A_{i}^{\perp}\cap\mathcal{M}^{n}_{\mathbb{C}}$ are pairwise
orthogonal.

Let $\mathcal{K}^{\perp}_{i}$ be the polar quadric of $\mathcal{K}_{i}$
relative to $\mathcal{M}^{n}_{\mathbb{C}}$. Each $\mathcal{K}^{\perp}_{i}$ is
a $(n-1)$-dimensional non-degenerate quadric that is contained in the
hyperplane $A_{i}^{\perp}$. By Lemma A.1, the centres of the hyperspheres in
$\overline{\mathbb{C}^{n}}$ of the $n+2$ different generations of
$\sigma_{\mathbb{C}}(\mathcal{D})$ are exactly the points of the quadrics
$\sigma_{\mathbb{C}}(\mathcal{K}^{\perp}_{i})$.

Let $N_{\mathbb{C}}^{\perp}$ be the tangent hyperplane of
$\mathcal{M}^{n}_{\mathbb{C}}$ at the point $N_{\mathbb{C}}$. The intersection
$N_{\mathbb{C}}^{\perp}\cap\mathcal{D}$ projects under $\sigma_{\mathbb{C}}$
to the quadric $\mathscr{C}^{n-2}$. The quadrics
$\sigma_{\mathbb{C}}(\mathcal{K}^{\perp}_{i})$ are confocal quadrics because
they belong to the dual of a pencil of quadrics such that $\mathscr{C}^{n-2}$
also belongs to the dual pencil. To see this, consider an arbitrary point in
$\mathscr{C}^{n-2}$. It can be described as the projection
$\sigma_{\mathbb{C}}(P)$ of some point $P\in
N_{\mathbb{C}}^{\perp}\cap\mathcal{D}$. Let $P^{\perp}$ be the polar
hyperplane relative to $\mathcal{M}_{\mathbb{C}}^{n}$. The point $P$ is
contained in all of the quadrics $\mathcal{K}_{i}$. Dually, $P^{\perp}$ is
tangent to all of the quadrics $\mathcal{K}_{i}^{\perp}$. The hyperplane
$P^{\perp}$ intersects $\overline{\mathbb{C}^{n}}$ in a $(n-1)$-dimensional
projective subspace that is tangent to $\mathscr{C}^{n-2}$ at the point where
the line $P\vee N_{\mathbb{C}}$ intersects $\mathscr{C}^{n-2}$. Each quadric
$\mathcal{K}^{\perp}_{i}$ is tangent to the hyperplane $P^{\perp}$. So, each
projected quadric $\sigma_{\mathbb{C}}(\mathcal{K}^{\perp}_{i})$ is tangent to
the foregoing $(n-1)$-dimensional projective subspace that is tangent to
$\mathscr{C}^{n-2}$. By Definition A.3, this means that the $n+2$ quadrics are
confocal. ∎

Figure 8 illustrates how cyclics can be generated according to Theorem A.4.

Figure 8: The envelope of the black circles is a cyclic. The generating
circles have a common orthogonal circle and their centres are contained in a
conic. Each generating circle is twice tangent to the cyclic. Some of the
pairs of tangency points are imaginary. Some of the generating circles are
imaginary. Any generating circle with null radius is a focal point of the
cyclic.

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Mathematics, Physics, Logic, Philosophy

A. N. Gorban

# Hilbert’s Sixth Problem:
the endless road to rigour

A. N. Gorban1 1Department of Mathematics, University of Leicester, Leicester
LE1 7RH, UK<EMAIL_ADDRESS>
###### Abstract

Introduction, where the essence of the Sixth Problem is discussed and the
content of this issue is introduced.

###### keywords:

Hilbert, axiomatizing, physics, quantum physics, continuum mechanics, kinetics

##  1 The Sixth Problem

In the year 1900 Hilbert presented his problems to the International Congress
of Mathematicians (he presented 10 problems at the talk; the full list of 23
problems was published later). The list of 23 Hilbert’s problems was very
influential for 20th century mathematics. The Sixth Problem concerns the
axiomatization of those parts of physics which are ready for a rigorous
mathematical approach. Hilbert’s original formulation (in English translation)
was:

“6. Mathematical Treatment of the Axioms of Physics. The investigations on the
foundations of geometry suggest the problem: To treat in the same manner, by
means of axioms, those physical sciences in which already today mathematics
plays an important part; in the first rank are the theory of probabilities and
mechanics.” This is definitely “a programmatic call” for the axiomatization of
existent physical theories.

In a further explanation Hilbert proposed two specific problems: (i) axiomatic
treatment of probability with limit theorems for the foundation of statistical
physics, and (ii) the rigorous theory of limiting processes “which lead from
the atomistic view to the laws of motion of continua”: "As to the axioms of
the theory of probabilities, it seems to me desirable that their logical
investigation should be accompanied by a rigorous and satisfactory development
of the method of mean values in mathematical physics, and in particular in the
kinetic theory of gases. … Boltzmann’s work on the principles of mechanics
suggests the problem of developing mathematically the limiting processes,
there merely indicated, which lead from the atomistic view to the laws of
motion of continua."

The Sixths Problem has inspired several waves of research. Its mathematical
content changes in time in a way that is very natural for a “programmatic
call” [1].

Figure 1: David Hilbert, the great reformer of science. The cover image of the
issue. Courtesy the artist Anna Gorban, 2018

In the 1930s, the axiomatic foundation of probability seemed to be finalized
on the basis of measure theory [2]. Nevertheless, Kolmogorov and Solomonoff in
the 1960s stimulated new interest in the foundation of probability
(Algorithmic Probability) [3].

Hilbert, Chapman, and Enskog created asymptotic expansions for the
hydrodynamic limit of the Boltzmann equations [4, 5]. The higher terms of the
Chapman-Enskog expansion are singular and truncation of this expansion does
not have rigorous sense (Bobylev [6]). Golse, Bardos, Levermore and Saint-
Raymond proved rigorously the Euler limit of the Boltzmann equation in the
scaling limit of very smooth flows [7, 8], but recently Slemrod used the exact
results of Karlin and Gorban [9, 10] and proposed a new Korteweg asymptotic of
the Boltzmann equation [11]. These works attracted much attention and a
special issue of BAMS was published with papers of L. Saint-Raymond [12], and
A.N. Gorban and I. Karlin [13]. There was a media reaction: QuantaMagazine
published on July 21, 2015 a paper “Famous Fluid Equations Are Incomplete: A
115-year effort to bridge the particle and fluid descriptions of nature has
led mathematicians to an unexpected answer.”

It seems that Hilbert presumed the kinetic level of description (the
“Boltzmann level”) as an intermediate step between the microscopic mechanical
description and the continuum mechanics. Nevertheless, this intermediate
description may be omitted. Now, L. Saint-Raymond with co-authors is
developing a new approach to the problem “from the atomistic view to the laws
of motion of continua” without intermediate kinetic equations [14].

Quantum mechanics was invented after the Hilbert problems were stated. Almost
simultaneously with the birth of quantum theory (in 1925), he devoted a
seminar to the description of its mathematical structure. The notes of this
seminar, collected by von Neumann and Nordheim (Hilbert’s assistants at that
time), were later published in a joint paper [15]. In this paper the authors
declare that the goal of an axiomatization of a physical discipline consists
in: “… formulating the physical requirements so clearly, that the mathematical
model becomes uniquely determined by them …”. The main idea is that a physical
theory consists of three, sharply distinguishable parts: (i) physical axioms,
(ii) analytic machinery (also called “formalism”), and (iii) physical
interpretation. The first attempt at formalization of quantum mechanics was
performed by von Neumann [16].

The Kolmogorov formalization of the classical probabilistic model was
published in a time (1933) when quantum mechanics was already making extensive
use of a completely different probabilistic formalism. In the following 20
years each of the two disciplines was strongly concentrated on its own inner
development and we had to wait until the second Berkeley Symposium on
Probability and Statistics (1950) when Feynman for the first time explained to
a large and qualified international audience of probabilists, the scientific
challenge posed by the existence of two, apparently mutually incompatible,
mathematical models of probability theory [17].

After that, the axiomatic approach to quantum probability was developed by
many researchers [18] and there are many versions of its axiomatization [19,
20]. The fast development of this area of research has involved ideas from
algebra, functional analysis, and fuzzy set theory. There has even developed a
programming language based on non-commutative logic [21]. Ideas and methods of
quantum computing [22, 23] and quantum cryptography [24] transform research in
the foundation of quantum mechanics into an applied discipline with a
perspective of applications in engineering. Many new mathematical structures
and methods have been invented (see, for example, [25, 26]). Applications of
quantum probability are now much wider than just the mechanics of subatomic
particles [27].

Work on Hilbert’s Sixth Problem involves many areas of mathematics:
mathematical logic, algebra, functional analysis, differential equations,
geometry, probability theory and random processes, theory of algorithms and
computational complexity, and many others. It remains one of the most seminal
area of interdisciplinary dialog in mathematics and mathematical physics.

##  2 This issue

In the first paper [28] L. Corry explains the essence of the Sixth Problem as
a programmatic call for the axiomatization of the physical sciences. Then two
reviews follow. R. Hudson [29] gives a survey of the “non-commutative” aspects
of quantum probability related to the Heisenberg commutation relation. L.
Accardi [30] explains that “One can say that, with the birth of quantum
theory, Hilbert’s Sixth Problem was split into three different questions:

  1. 1.

Axiomatize classical probalility.

  2. 2.

Axiomatize the new (quantum) probalility.

  3. 3.

Clarify the connections between the two.”

He presents systematically the Kolmogorov compatibility conditions, the
differences between Non-Kolmogorovian and Kolmogorovian models, and the
information theoretic formulation of Heisenberg principle. Emergence of
Hilbert spaces, the Schrödinger equations, and gauge theories is demonstrated.

Lozada Aguilar, Khrennikov, and Oleschko in their ‘opinion piece’ paper [31]
move the discussion much further and present a nonclassical application of
nonclassical probability. They go from axiomatics of quantum probability and
the theory of open quantum systems to modelling of geological uncertainty and
management of intelligent hydrocarbon reservoirs.

F. Golse [32] develops a bridge between the quantum and classical parts of
Hilbert’s Sixth Problem: he studies the semiclassical mean-field limit for the
quantum N-body problem and finds a convergence rate for the mean-field limit.
This work is closely related to the analysis of “the limiting processes, there
merely indicated, which lead from the atomistic view to the laws of motion of
continua”, requested by Hilbert.

S. Majid in his opinion piece [33] goes beyond the commonly accepted quantum
theories. The mission of his approach is to be a step towards understanding
“how both General Relativity and Quantum Theory could emerge from some deeper
theory of quantum gravity”. He also reviews some previous works and proposes a
simple toy 4-point model, which aims to give a hint about quantum gravity in
space-time.

The series of works devoted to the “satisfactory development of the method of
mean values in mathematical physics, and in particular in the kinetic theory
of gases” is opened by the paper of A. Bobylev [34]. He considers the problem
of higher equations of hydrodynamics. For this purpose, he proposes and
analyses the method of successive changes of hydrodynamic variables and
estimates the accuracy of the Navier-Stokes and higher approximations. The
problem of the proper reduction from kinetics to fluid dynamics, stated by
Hilbert in 1900, is not yet solved. New equations of continuum mechanics
between Navier–Stokes and Boltzmann equations may be needed. M. Slemrod [35]
states that the classical “Boltzmann to Euler limit” fails in the most
interesting cases, and the equations should be changed. He uses as the
prototype the exactly solved reduction problem for simplified kinetics [9, 13]
and proposes the Korteweg fluid dynamics for the non-equilibrium flows. L.G.
Margolin [36] also aims to derive new fluid dynamic equations. He uses the
local averaging in the coordinate space and derives the hydrodynamics models
for the averaged moments. I. Karlin [37] demonstrates how to apply the regular
principle of dynamic invariance for regularisation of Grad’s moment system. He
finds and deciphers the detailed structure of the regularised 13 moment
system. R. MacKay [38] considers the reduction problem for Markov processes in
more formal mathematical settings and proposes a new procedure for such
reduction.

A.N. Gorban and I.Y. Tyukin [39] show how the programmatic call of Hilbert’s
Sixth problem influences statistical mechanics and machine learning. They
present new aspects of the concentration of measure phenomena, the stochastic
separation theorems, and new non-destructive procedures for correction of
unavoidable errors for artificial intelligence systems.

The free style opinion discussion of a possible full mathematisation of
physical theories and a novel algorithmic paradigm for physics is presented by
G.M. D’Ariano [40].

##  3 The problem of reality: what we keep silence about

Semantics of physics should be different both from the model-theoretical
semantics of formal theories and from the various linguistic and epistemic
semantics. Development of semantic of physical reality is a non-trivial task:
it should have sufficient level or rigour for use in theoretical physics and
sufficient level of generality for work with the wide range of physical
theories, including yet unknown future physics.

Here we briefly observe, from a bird’s eye view, the problem of semantics
associated with the Sixth Problem. For discussion of semantic we employ the
idea of possible worlds in its wide sense. The possible world of a theory is a
possible course of events, with full details, which is fully concordant with
the theory.

Because of our education, our minds live in a bizarre configuration of
possible worlds of physical theories. Mechanics (with continuum mechanics),
electromagnetism, and optics cover our everyday experience. If we meet
anything different, we refer to other physical disciplines: statistical
physics, quantum mechanics, and so on.

The idea that there exists a general theory, the truth, which rules the world,
is a very powerful driving force for theoretical physics. Einstein, for most
of his life, tried to approach such a theory. He definitely knew that the
existence of this theory cannot be proved but it can just be postulated. In
his famous discussion with R. Tagor [43], Einstein formulated his point of
view unambiguously:

  * EINSTEIN: I cannot prove, but I believe in the Pythagorean argument, that the truth is independent of human beings. It is the problem of the logic of continuity.

  * ……………………..

  * TAGORE: In any case, if there be any truth absolutely unrelated to humanity, then for us it is absolutely non-existing.

  * EINSTEIN: Then I am more religious than you are!

It is absolutely necessary to mention here for clarity that Einstein often
expressed scepticism with regard to the traditional creeds and God for him was
similar to the God of Baruch Spinoza. (His sentence “I believe in Spinoza’s
God, who reveals himself in the harmony of all that exists, …” became well-
known [45]). For Spinoza, God was natura naturans (nature doing what nature
does) [44].

We can call the belief in the rational construction behind the Universe, the
ultimate rationalism. The idea of “ultimate rationalism which urges forward
science and philosophy alike” belongs to Whitehead [46].

I propose to take seriously the title of the Sixth Problem: “Mathematical
Treatment of the Axioms of Physics”. Of course, the explanation that follows
the headline, reduces the sublime sound of the title. But we should take into
account that Hilbert was a highly qualified scientist and an experienced
science writer. He understood the potential reaction of the readers of this
title, no doubt. This could not be done unintentionally.

If we consider this problem seriously then we should be ready to assume that
the real Universe is a possible world of a theory (yet unknown) and our goal
is to reveal this theory. Einstein insisted that the proper theory should be
categorical as much as it is possible: ideally, the structure described by the
theory should be practically unique. (He discussed relations between the
equations, which are fixed by the theory, and the initial conditions, which
are arbitrary, and proposed to decrease this freedom of choice by additional
theoretical principles like the Mach principle.)

Alas, it is quite possible that we aim to create the “theory of everything”
but instead of a beautiful theory always approach a system of “standard
models”, which do not pretend to be the truth and, possibly, have a hidden
contradiction, either in the theory, or with the reality. It may happen that
instead of the possible world of a nice theory, our world will be always an
“impossible possible world” with hidden contradictions [47].

The modal and epistemic logics consider the contradictions and impossible
worlds as epistemic problems, whereas in ‘reality’ there exists the truth and
non-contradictory semantics. Here we allow another possibility: there is no
non-contradictory theory of everything (even hidden), and the world has no
rational reconstruction. We also cannot exclude the possibility that the
rational truth is so well-hidden that we cannot reach it. The apparent result
should be the same: standard models and hidden contradictions.

The web of standard models, which contradict each other and have known
mismatches with reality, is the everyday practice of applied science. The
ultimate rationalism is a driver that moves us to new, beautiful and simple
theories. Nobody can state that it will release us from the web of standard
models and from the mousetraps of impossible possible worlds, but the dream
helps and gives energy.

Another problem with the semantics of physics is that there exists no
atheoretical formalisation of a notion of physical reality. The possible
worlds of theoretical physics consist of things, their attributes and
relations prescribed by the theory. We have no semantics of physics without
physics. The proposal of D’Ariano to discuss “physics without physics” [40] is
not just an elegant expression. We need the universal semantics. The attempts
from the beginning of the Twentieth Century to exclude theoretical terms and
work with the language that reflects the everyday human experience are a bit
naïve. Everybody who has experience in robotic vision knows that it is a
highly non-trivial task to teach a robot the everyday experience of scene
analysis and pattern recognition. Everyday natural language includes
constructs, which are also far from direct perception.

We can try to outline the possible “semantic of physical reality” that
complements semantics of physical theories, is simple, abstract, and free from
theoretical allusions. Let us just follow the machine learning abstractions.
There are two alphabets: Perceptions $\mathcal{P}$ and Actions $\mathcal{A}$.
The “person” selects and sends “outwards” the symbols from $\mathcal{A}$
(actions) and receives symbols from $\mathcal{P}$ (the answers to actions, or
just signals from “outsides”). There is the “freedom of will”: selection of a
symbol from $\mathcal{A}$ has no restrictions ($\mathcal{A}$ models commands
to actuators, the real actions and their results return to the person with the
$\mathcal{P}$-feedback).

The outer world can be modelled in any theory, can act as a classical or
quantum device, etc. The person can learn, explore the world, create theories
and identify models. At the end, the sets of $\mathcal{A}-\mathcal{P}$
sequences will correspond to sets of structures and models.

We should not expect that such learning is driven by a utility optimisation.
Quite the opposite, this learning can be goal-free. Gromov [48] identified
structure = interesting structure and declared that the goal free structure
learning is a structurally interesting process.

This is just a brief outline of the semantic problem related to the Sixth
Problem:

  * •

The problem of the ultimate rationalism versus the web of standard models and
the possible impossible worlds;

  * •

The problem of a semantics of reality, which does not depend on a theory and
is not related to human physiology or natural language. The approach
considered is borrowed from machine learning;

  * •

The problem of structure discovery, model identification and goal-free
learning.

##  4 Conclusion

Hilbert started his “road to formal rigour” from “The Foundations of Geometry”
[41]. He expected, that in an axiomatic form, the theory would be developed
independently of any need for intuition: “Professor Hilbert has, so to speak,
sought to put the axioms into such a form that they might be applied by a
person who would not understand their meaning because he had never seen either
point or straight line or plane” [42].

In 1900, Hilbert’s axiomatizing project was well accepted by the community of
mathematicians. Some minor comments about incompleteness of his “Foundations”
required just incremental work. Gödel was not yet born. Hilbert’s Sixth
Problem was the declaration of the expansion of the axiomatic method outside
the existing mathematical disciplines, in physics and beyond.

There are three general lessons from the history of the Sixth Problem:

  * •

Attempts at axiomatizing are useful even beyond mathematics: they help to
clarify the basic assumptions and keep the mind in order. This is good and
encouraging.

  * •

This order may be much more complicated and counter-intuitive than the
preceding chaos. This is not so encouraging, but not too bad: axiomatizing is
a powerful tool, and its use requires accuracy.

  * •

The road to rigour is infinite. This is neither encouraging nor discouraging:
this is reality.

Discussions of Hilbert’s Sixth Problem and preparation of this issue gave us
one more lesson: The road to rigour is infinite and interesting. Some comments
by the Workshop participants are collected after the bibliography.

##  Appendix A Opinions of workshop participants

“This workshop was rather unique in its kind as it brought together
mathematicians from very different horizons (from probability theory to
statistical physics, partial differential equations, logic and complexity),
whose common interest lies in Hilbert’s sixth problem. The statement of
Hilbert’s sixth problem is indeed so broad that it can be studied from a
multitude of angles, and it was really extremely interesting for me to hear
some of the best specialists from those various fields explain their viewpoint
of the problem, and the progress that has been made recently. I think all the
participants share this impression on having gone on a wonderful tour of the
sixth problem for three days, and have come home with new ideas, and lots of
new questions.

Isabelle Gallagher

Université Paris-Diderot (Paris 7),

UFR de Mathématiques,

Paris, France.

“As a historian of mathematics who has devoted many years of his academic life
to research the origins, the development and the overall impact of Hilbert’s
sixth problem, this conference has been for me a unique kind of scholarly
experience. Historians of mathematics are used to work in intellectual
isolation. Typically, our work is too technical and daunting for historians of
science in general. At the same our work is too “historical” for what should
be one of our main natural audiences, namely, that of the mathematicians (and
in the case of Hilbert’s sixth problem, also the physicists). I have had
previous opportunities to speak to mathematical audiences about my work on the
problem, but with special attention to the encounter between Hilbert and
Einstein around the final formulation of the field equations of general
relativity. Quite unsurprisingly, this has proved to be an appealing enough
topic for such audiences. But this conference was fully devoted to the much
broader topics associated with the Hilbert problem, and to a truly wide range
of topics, and it was very refreshing to realize the extent to which the ideas
put forward by Hilbert more than 115 years proved to be much more relevant and
of long-term impact than Hilbert himself could have ever conceived when
formulating his programmatic call for the axiomatization of physical theories
in 1900. The talks were not only informative but also inspiring, and for me
they were an eye opener in many directions. I was also glad to come out with
the feeling that my own historical talk arose significant interest among the
attendees.

Leo Corry,

The Lester and Sally Entin Faculty of Humanities,

Tel Aviv University

Israel

“Hilbert was one of the most successful leaders to set foundations of
mathematics by axiomatic methods. Hilbert’s sixth problem called for extending
this method out of mathematics to physical science. Mathematicians who made
significant contributions to this problem in the last decades world wide
gathered and discussed the significant roles of the problem playing mainly in
hydromechanics and quantum physics. The workshop revealed the unexhaustive
value of the problem, which has enriched both physical science and mathematics
for more than 100 years and will lead to unexpected mutual interactions
between those intellectual arts.

Masanao Ozawa

Graduate School of Information Science

Nagoya University

Japan

“The Workshop on “Hilbert’s Sixth Problem” was an inspriring experience that
shed a new light on the problem itself and on the approaches of its solution.
The carefully selected lectures from both the classical and quantum problems
of mathematical physics have clearly showed that Hilbert’s Sixth Problem has
not been completely solved but presents a lot of interesting research
questions to be answered even today not only in applied mathematics and
physics but also in engineering. Thanks to the excellent organization and to
the friendly and relaxed atmosphere there were plenty of opportunities for
discussions and hopefully cross-fertilization of the ideas and solution
methods in the interdisciplinary area between applied mathematics, physics and
a bit of engineering.

Katalin Hangos

Process Control Research Group

Computer and Automation Research Institute

Hungarian Academy of Sciences

Budapest, Hungary

“In 1900, the German mathematician David Hilbert laid out a program (known as
Hilbert’s 6th problem) calling for no less than the “axiomatization of
physics”. As an example, Hilbert proposed to derive the classical equations of
fluid mechanics from the atomistic description of gases — at a time when even
the existence of atoms was a matter of heated scientific debate. Since then,
new paradigms have emerged in physics: relativity, and quantum mechanics have
altered our understanding of the world. Today, the quest for a mathematical
coherence in modern physics is even more formidable a challenge than in the
days of Hilbert. The workshop organized by Prof. A. Gorban and his colleagues
at the University of Leicester has been a unique opportunity to assemble a
panel of mathematical physicists from various countries (including for
instance China, France, Italy, Russia, Sweden, Switzerland, in addition to the
UK), with a great variety of viewpoints on Hilbert’s 6th problem.

François Golse

Département de Mathématiques

École Polytechnique,

Paris, France

“The Workshop on “Hilbert’s Sixth Problem” organized by Alexander Gorban in
Leicester will have a great impact on the community of physicists and
mathematicians. For the first time a workshop has been devoted to a problem of
the outmost relevance at the methodological level for the advancement of
theoretical physics, in a time of great evolution of theoretical physics.
Indeed, the solution of the VI Hilbert problem, namely the axiomatization of
physics, will play a pivotal role in finding the correct answer to the major
problems in contemporary physics, including the solution of the logical clash
between General Relativity and Quantum Theory, and that of the emergence of
Hydrodynamics from Statistical Mechanics. We have attended very interesting
talks during this conference, and we have come back home with a todo list of
relevant issues to address and a set of methodological recipes for improve the
quality of theoretical research. David Hilbert has changed the history of
physics with his research and his teaching in Göttingen during the quantum and
relativity revolution, and he is still teaching to us nowadays the most
important lesson: that the long-lasting relevant science must be built on
solid foundations.

Giacomo Mauro D’Ariano

Istituto Nazionale di Fisica della Materia,

Unita’ di Pavia,

Pavia, Italy

“In 1900 David Hilbert provided the mathematicians of his day with a list of
problems for the twentieth century and in particular his sixth problem which
called for the axiomatization of physics in the same spirit as we would treat
geometry. The problem in its various interpretations has provided a fertile
ground for mathematicians for over one hundred years with a rather impressive
list of results and even unintended consequences. This conference (perhaps the
first of its kind) expanded on the physics known to Hilbert in 1900
(statistical and continuum mechanics) and included the issues of the very
small (quantum mechanics), very large (general relativity), and basic
fundamental issues of probability which is the key component of both the
statistical and quantum view of nature. At first thought it seems like a
workshop with this broad view would have the difficulty of too large a
perspective but the just the reverse was true. The participants excelled at
thinking about the big picture and their presentations and conversations went
far beyond any possible narrowness and self-interestedness. It was a
remarkable event.

Marshall Slemrod

Dept of Mathematics,

University of Wisconsin,

Madison, WI USA

The author declares that he has no competing interests.

The Workshop “Hilbert’s Sixth Problem” (University of Leicester, May 02-04,
2016) and preparation of this issue were supported by EPSRC grant
EP/N022653/1, LMS conference grant No: 11503, and IMA Small Grant SGS47_15.

The idea of this issue was proposed during the LMS-EPSRC-IMA supported
workshop (conference) “Hilbert’s Sixth Problem” (Leicester, May 2016)
organised by Alexander Gorban. The Scientific Committee of the Workshop was:
Luigi Accardi, Alexander Bobylev, Pierre Degond, and Marshall Slemrod. Most of
the papers in the issue grow from the talks at the Workshop. We are very
grateful to all the participants for the inspiring talks and seminal
discussion!

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