SPACE AND SHAPE

                                Pat Hayes

Spatial objects are things which occupy some piece of space (contrast
integers and timeintervals). They typically have many properties which
aren't spatial, but the purely spatial aspects of them can be divided into
three kinds, all of which can change independently of the others. Not all
spatial objects have all these properties and some may have others in
addition.

1. The intrinsic shape of the object.
2. The location of the object in some larger reference space or region.
3. The orientation of the object in some larger reference space, or with
respect to something else.

In order to be able to talk about orientation, we will adapt the notion of
'intrinsic shape' to include a set of intrinsic axes. So for example a
cylinder with its main axis along the cylinder will be considered a
different shape than the same cylinder with its main axis perpendicular to
the cylinder. This requires us to split the notion of 'shape' into two
distinct notions:

A directed shape is a shape with between one and three mutually
perpendicular axes 'rigidly embedded' in it. Particular examples include a
directed surface, which has one axis (locally) perpendicular to the surface
and two axes in the surface, and the shape of a Ford Thunderbird with axes
in the forward, upward and left-to-right directions. (Exactly what is meant
by a 'shape' and an 'axis' is given in more detail later. ) One key property
is that moving or rotating an object does not change the directed shape, but
other transformations (folding, bending, compression, twisting, forced
conformity to another object, impacts, etc.) may change the shape. In
general, an operation which changes shape is classed as a deformation. If a
directed shape has more than one axis then the axes are ordered as primary,
secondary and tertiary, conventionally labelled by the letters a,b,c
respectively. They always form a right-handed set (if one were standing
facing the primary axis with the secondary axis vertical, then the tertiary
axis is on the right.)

A spatial extent is the piece of space actually occupied by an object, ie
the piece of space which is filled up with the material which comprises the
object. This is supposed to be defined relative to some larger reference
space, so such pieces of space do not move when the object is moved (though
they may move when the larger reference space is moved.) Any directed shape
in a given orientation (ie with its axes fixed in affine 3-space) and at a
given location, occupies a unique spatial extent, but two different objects
can have the same spatial extent (though not at the same time).

A directed shape is symmetric about an axis if it occupies the same spatial
extent when rotated about that axis.

Shapes and axes

We define shape in terms of planar approximations. (Smooth shapes can be
regarded as limits of this approach, but for RKF textbook applications the
distinction isnt likely to be important.) The basic notion is the simplest
kind of directed shape, a directed surface.

A directed surface (edge) is a roughly planar (linear) piece of surface with
a unit vector perpendicular to its tangent(s). Another name for this is a
side. Each piece of 2Dsurface defines two directed surfaces which have their
unit vectors in opposite directions. Intuitively, a directed surface is one
side of an abstract piece of 2D space. (Each piece of 1-D surface (ie a
line) defines two directed edges in any containing plane similarly, but
there are infinitely many directed edges surrounding each edge in 3space.)
Basic facts about directed surfaces are that they come in pairs, have
opposites, and that surfaces of solid things are always directed outwards.
For example, a cell membrane consists of two opposite sheets each of which
has a hydrophilic side and a hydrophobic one, which are in sheetlike contact
across their hydrophobic directed surfaces. So the membrane has two outward
directed surfaces which are both hydrophilic.

(Another way to think about a directed surface is as a very thin
surface-like region of space immediately adjacent to the surface on one side
of it. If I paint a directed surface, this region gets filled with paint.
Mathematically this region is the product of the surface with its first
spatial differential in the direction of the vector. This is the sense of
'directed surface' I used in the 'liquids' axioms)

A shape is a set of pieces of directed surface (or directed edge) in a
specified 3-dimensional (or 2-dimensional) arrangment. Idea is that one can
move a shape around and its still the same shape, but if you move its parts
relative to each other then it becomes a different shape. Particular shapes
include eg a facial mask of Adolf Hitler, the shape of the letter A, and a
particular configuration of an Alexander Calder mobile. Often we will not
need to describe a shape in detail, only know that it has some global
properties and bears some useful relation to another shape. Examples of
global properties of a shape include contiguity (theres a path in the
surface from any part to any other part) smoothness, roughness, angularity
and so on, and flat where all the pieces are coplanar; spherical, etc..

A key relationship between shapes is opposite, gotten by reversing the
directions of all the directed surfaces and edges in the shape. Two shapes
which are opposite can fit perfectly against each other so that each part of
one shape is in contact with a part of the other and vice versa. A key fits
part of the interior of a lock in this sense.

This concept of 'fitting' arises in the idea of 'conformity': one
surface/object conforms to another if it takes its opposite shape when they
are forced into contact. This is the essential core of the the soft/hard
contrast: when you push 2 things together, the soft one distorts to fit the
hard one. Examples include wrapping a statue in clingwrap, pouring paint
onto a surface, and pushing clay into a mold. (May need to generalise this
to a degree of fit, ie more pieces of surface are in contact than not. Good
fit provides more contact places than poor fit. Perfect fit has them all.
Example of perfect fit is liquid against a surface, as in making a casting
from a mold, or making a cast of a face.)

An axis or direction vector is a vector which has a direction but no
particular location or length (it is a unit vector in affine space). This
means that it does not make sense to ask where in a directed shape its axes
are located (through the middle or along the edge, etc.); they define only a
direction and nothing else.

The set of all such directions forms a mathematical group with several
generators, including normalised vector addition, rotations and reflection.
The inverse operation in this group is called opposite. There is a tractable
subgroup generated by clockwise pi/2 rotations about an axis plus
reflections, and this can be given a neat axiomatisation. This 'right-angle'
group is sufficient to describe notions of parallelism, perpendicularity and
some kinds of symmetry. Using this we should be able to describe things like
perpendicular attachement, lying parallel to a surface, rotation, and
motions such as sliding, rotating, and tumbling.

Suppose opposite directions are denoted by the function opp and CR(x,y, z)
means a pi/2 clockwise rotation about x transforms y into (what was) z.
Anticlockwise rotation is got by applying opp to the last argument. The
axioms are:
(opp (opp x)) = x
(implies CR(x,y, z) (not (x=y or x=z or y=z)) )
(implies CR(x,y,z) CR(x, z, opp(y)))
(implies CR(x,y,z) CR(y, x, z))
(implies CR(x,y,z) CR(z, y, x))
(and maybe some more which I will try to get straight soon.)
It follows from these for example that a pi rotation sends the other two
axes to their opposites, and that a 2pi rotation is the identity function,
and that if you face north, lie on your back, turn clockwise by pi/2 and
then stand up, you will be facing west.

Some suggestions for assigning primary axes to shapes (all pretty obvious):
cylinder or rod - along axis of cylinder
plane - perpendicular to plane
hexalateral object - backtofrontaxis
blob - through object perpendicular to main side (ie use axis of main side
as a surface)

Extents and locations and pieces of space.

We think of space as basically consisting of a collection of locations where
something can be placed. These locations are defined with respect to a
larger reference space which contains them all, called a region. Regions may
themselves be places in some larger region, and locations or objects located
in a region may themselves define other regions for smaller locations. For
example, the back right passenger seat may be a location in a region
consisting of the interior of a car, which is itself located in geographical
space on the surface of the planet, which is moving within the solar system.

Exactly what counts as a location is left open, but it should be precise
enough to count as an answer to a question of the form 'where is x?'.
Different notions of precision may be suitable in different contexts. We can
allow only three-dimensional locations, or can allow things like 2d surfaces
and lines and even points as locations. Locations may overlap and be
included in one another, and may be hollow or arbitrarily shaped; or we may
impose a coarse grid-like discipline on a set of locations in a region if
that is more suitable. One basic relation between locations or pieces of
space generally is 'inside', meaning that one location is entirely included
in the other, and 'adjacent', meaning that the two pieces of space are
immediately adjacent with no space between them. There are several
alternative axiomatisations of thse notions already available, and we can
probably get by with a fairly simple algebraic version like RCC8. Motion of
an object (within a region) is conceptualised as change of its location in
that region; notice that the region may itself be moving, so whether or not
a thing is moving depends on the region (eg one can be moving at hundreds of
miles an hour while asleep in an airplane seat).

(This ignores such issues as whether the reference region is an inertial
frame, whether or not it has a gravity vector, and so on. Many of the Cyc
concepts seem to have an implicit assumption that the reference frame is
geographical space, but we cannot make this assumption for RKF.)

An extent is the most precise version of a location. The extent of a spatial
object is the precise piece of space occupied by the stuff which comprises
the object. For example, the extent of an ocean is the space completely
filled with water (plus fish and other detria), while the extent of a
plastic coffee cup is a thin, concave volume entirely filled with solid
plastic. Extents may have a complicated structure; in particular they may
have holes, cavities, etc, and may be not simply connected. Like locations
in general, extents do not move: when an object moves it leaves one extent
behind and fills up a new one. That is, we think of objects as moving
through the larger space in which they move; but if they define a region (eg
the space inside the empty cup) then that region moves with them. A
prototypical example of this is a container, which when moved takes its
contents with it. Relative to the container they remain in the same place,
but that place itself moves along with the container.

We need extents to be able to talk about rotational symmetry. An object is
symmetric with respect to an axis if it occupies the same extent when
rotated about that axis. For example, a cylinder with a longitudinal axis is
symmetric wrt that axis. (Limited forms of symmetry can be described by the
CR group, which cannot distinguish between the symmetry of a cube and that
of a sphere. Neverthless I suspect this will be good enough for most of the
cellular chemistry reasoning.)

Assemblies

Spatial things can often be made up from assemblies of other spatial things.
A general thoery of this wojuld encompass all of structural engineering and
solid geometry, but we can make some useful basic descriptions based on some
simple shapes. For example consider the following shape classifications:
rod
ribbon
spiral
tube
2Dsheet
3Dblob
These can be assembled together in a variety of ways.. All the following
arise in protein folding and membrane structure for example:

A rod can twist into a spiral
A spiral can be a tube
Several parallel rods can make a tube (with the same axis and length as the
rods)
A folded ribbon can make a blob (with same width as ribbon but much shorter)

A folded rod or ribbon can make a 2Dsheet (like weaving)
A 2Dsheet can be rolled into a tube
A series of pieces can be linked by hinged or flexible joints end to end to
form a foldable structure
A tube can pass through a 2Dsheet (ie be embedded in the sheet with its axis
perpendicular to the sheet and each end in contact with the space on either
side of the sheet.)
A foldable structure can be woven through a sheet from one side to the other
and back again.

Detailed descriptions of these various possibilities can be given using the
concpets developed earlier. For example, a hinge is a shared linear edge
about which the two pieces can rotate. If they begin coplanar then a pi
rotation about the hinge will bring their original parallel faces into
opposite orientation and adjacent. If they fit, then many pieces of these
directed surfaces will be exactly opposite and adjacent, so any attraction
between them will be reinforced (we here need an axiom about how many small
forces can add up to a strong force, which may be tricky to state!) and this
position will be stable. If this is repeated, many layers will be added to
form a layered structure consisting of a 3dblob made of many 2dsheets.

We are currently formalising a catalog of various geometric assembly and
deformation processes which can be used to describe the cellular structures
and processes in the text.

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