oct 14    DRAFT SUBJECT TO CORRECTION AND CHANGE

Formal conclusions depend on formal assumptions. This paper sets out new assumptions for constructing mathematical models from physical models that differ from those now implicitly used. Using the new assumptions, derivations of differential equations from coupled physical models change. Terms that are infinitessimal according to current procedures are finite. The new conclusions seem justified by logic and evidence.

The idea that mathematics can be an imperfect and obscure representation of physical circumstances is accepted.

"As far as the laws of mathematics refer to reality, they are not certain; as far as they are certain, they do not refer to reality." Albert Einstein

Indeed, Einstein and others have expressed surprise that mathematical equations represent reality at all, and have found that representation a deep mystery. Many seem to have lost hope of a more "rigorous" mapping between physical reality and mathematics(1). At the same time, the IMPORTANCE of accurate and conceptually clear mapping has become more and more accepted. According to George Johnson

" Scientists must constantly remind themselves that the map
is not the territory, that the models might not be capturing
the essence of the problem, and that the assumptions built
into a simulation might be wrong.(2) "

How might mathematical mapping more closely approach a cartographic ideal? The fundamental ideal of cartography seems to be congruence - when maps are made of territories, the map must represent the details of the territory that are of interest, in a way that permits checking, back and forth, between the map and the territory. If there is a character or pattern on the map, it should be clear what in the territory it refers to, and it should convey the information about the territory that a user needs and expects.

Often in cartography, there are maps of maps. The more detailed maps are unweildy, and so more condensed maps are needed. But for the more condensed maps to be constructed, they must refer to the more detailed maps - not directly to the complexities of the territory.

I suggest, with S.J. Kline, that in order to model physical circumstances into mathematics, we need an intermediate map, capable of representing details that more abstract mathematical formulations cannot. I suggest that representations in this intermediate map must be clear before mapping into more abstract mathematical domains can validly be done. Motivation for this suggestion comes from neurophysiology, where the standard neural conduction equation seems to be inconsistent with data. Derivation of a neural conduction equation according to the mapping scheme below seems to fit data much better.

The natural world involves details of geometry, dimension, measurement procedure, and physical law that abstract mathematics does not encompass or clearly represent. We can map to natural laws because there are many linear natural law operators(3), commonly called dimensional parameters, that have been almost (but not quite) trouble free. Linear natural law operators (dimensional parameters) encode relations (including physical dimensions, physical laws, and measurement procedures) from an EXTRAMATHEMATICAL world into mathematics. We need an intermediate mathematical mapping domain that can accomodate these linear natural law operators. Let's call this domain the concrete domain of the measurable. We describe what we want concrete domain of the measurable to do, and the axioms and rules on which it is based, as follows.

Because we want map-territory correspondence in the cartographic representation sense, we want representation in the concrete domain of the measurable to have

algebra that is consistent with itself, and that fits the constraints of

equation-model geometry correspondence

and

dimensional consistency.

We want to be able to insist on these constencies in every term of an equation before that equation in the concrete domain of the measurable can be considered to be defined.

We want algebra in the concrete domain of the measurable to be ARITHMETICALLY CONSISTENT from one unit system to another.

Our current modeling does not meet these requirements. (Appendix **). When we construct model equations from coupled physical models according to the current procedures, we have terms that are not clearly related to the geometry of the physical model. These terms are not defined in a dimensionally consistent way. These terms, in the conventionally written finite form, are also arithmetically inconsistent from one unit system to another.

Hart has shown persuasively that dimensional calculations cannot be validly done within the dimensionless rules of the real numbers, but must occur, explicitly or implicitly, with dimensioned scalers(4). He has set out a Typed Family of Fields (TFF) formulation to show important, long neglected aspects of dimensional calculation, particularly those involve with linear algebra. But before a model based on a physical circumstance can be mapped into Hart's TFF formulation, it must be mapped in such a way that geometrical, dimensional, and arithmetical consistency of terms can be assured. A mapping into a concrete domain of the measurable according to the following assumption-axioms does this.

ASSUMPTIONS FIT TO CONCRETE, MEASURABLE MODELING THAT CAN INTERFACE WITH ABSTRACT MATHEMATICAL DOMAINS.

The following definitions and axioms appear to make step-by-step sense of modeling from the physical-measurable to the abstract mathematical. The requirements of the concrete and the abstract are not the same.

definition: CONCRETE: considered in direct and rule-bound connection to a particular object or a particular class of objects under specifically described circumstances: specifically connected to embodiment.

definition: ABSTRACT: considered apart from any application to a particular object or specific instance: separated from embodiment.

There must logically be two kinds of domains.

First, concrete domains of the measurable must exist that accommodate algebra, but that also include full details of system specification and operational measurement definition that permit physical laws related to specific, concrete circumstances to be expressed in terms that include linear natural law operators, relevant dimensions, and system variables.

Second, an abstract domain must exist, where entirely abstracted symbols (real numbers or dimensioned scalars) are manipulated.

We must be able to start from a concrete domain of the measurable, map results into the abstract domain, do abstract mathematical operations in the abstract domain, and then relate the results of these abstract operations back into a concrete domain of the measurable. We must be able to take these steps without losing information of concrete importance.

definition:    A linear natural law operator (also known as a dimensional parameter) is a dimensional transform ratio number that relates two measurable functions numerically and dimensionally. The linear natural law operator is defined by measurements (or "hypothetical measurements") of two related measurable functions A and B. The linear natural law operator is the algebraically simplified expression of {A/B} as defined in   A = {{A/B}} B.   The linear natural law operator is a transform relation from one dimensional system to another.  The linear natural law operator is also a numerical constant of proportionality between A and B (a parameter of the system.)   The linear natural law operator is valid within specific domains of definition of A and B that are operationally defined by measurement procedures.

Resistance per unit length is an example of a linear natural law operator. Resistance per unit length of a wire, R, is the ratio of voltage drop per length of that wire to current flow over that length of wire.    A resistance per unit length determined for a specific wire for ONE specific length increment and ONE specific current works for an INFINITE SET of other length increments and currents on that wire (holding temperature the same, and assuming wire homogeneity.)    R is the linear natural law operator in Ohm's law for a wire.    Other physical laws have other linear natural law operators.  The linear natural law operators (dimensional parameters) encode experimental information into a compact, linear form.

Here are some linear natural law operators (dimensional parameters) in common use:

mass, density, viscosity, bulk modulus, thermal conductivity, thermal diffusivity, resistance (lumped), resistance (per unit length) inductance (per unit length), magnetic susceptibility, emittance, ionization potential, reluctance, coefficient of restitution, and many more.

These are not axiomatic constructs, but they are essential in the modeling of physical laws.

Terms that express physical laws have a particular structure that relates a linear natural law operator, a space-time geometry, and a system variable. Newton's law of inertia, Ohm's law, applied to a lump resistor, and Ohm's law applied along a wire, offer examples of that structure. These equations make sense in reference to specific physical situations with specific geometries, and must be expressed in consistent unit system. The linear natural law operators m, Rp and Rl may be considered as algebraically simplified expression of

{f/a} ; {v1-2/i} ; and { (v(x+x.t)-v(x,t))/i }

for specific values of these dimensional variables that apply GENERALLY within a domain of definition. It is the generality of the linear natural law operators which makes them more than algebraic entities. It is this generality that makes it possible for us to describe physical laws algebraically. On the right hand side of () and () there is a natural law operator defined at a point or between fixed references, times an independent variable. The product determines a resultant variable on the left side. On the right hand side of () there is a natural law operator defined per unit length, associated with a specific length, times an independent variable. The product determines a resultant variable on the left side.

Here are some simple axioms, that permit us to define physical model equations completely in a concrete domain of the measurable. After this complete definition is done, we may readily map these equations into the domain of the algebra or a domain of dimensioned scalars. With these axioms, we can define the terms Maxwell found "difficult to interpret."

AXIOMS INVOLVING LINEAR NATURAL LAW OPERATORS (DIMENSIONAL PARAMETERS) and the GEOMETRY of CONCRETE EQUATIONS.

A1) Axiom: The axiom of geometrical consistency. When an equation refers to a particular geometry, then EVERY TERM must refer to that particular geometry. If the equation refers to an increment of volume (length, area, time), EVERY TERM must contain an increment of volume (length, area, time) and a linear natural law operator expressed per unit volume (length, area, time).

A2) Axiom: The axiom of dimensional consistency. Every variable and natural law operator in the equation must be defined in a consistent system of units that may, with sufficiently careful and complete specification, be reduced to units of length, mass, and time.

a21 Every variable in the equation must be set out in that same system of units.

a22 Every linear natural law operator in the equation must be in that same system of units. This means that every linear natural law operator must be expressible as a numerical value times a unit group (reducible to L,M,T or units traceable to L, M, T) where ALL lengths (explicit or implicit) are the same unit of length, ALL masses (explicit or implicit) are the same unit of mass, and ALL times (explicit or implicit) are the same unit of time.

A3) Axiom: CONSTRUCTED LINEAR NATURAL LAW OPERATORS: When two linear natural law operators (dimensional parameters) occur in product or ratio form as a result of a valid algebraic construction in the concrete measurable domain where they are defined, these dimensional parameters can be algebraically simplified together. Numerical parts are multiplied (divided) together. Unit exponents are added (subtracted.) The result is a CONSTRUCTED LINEAR NATURAL LAW OPERATOR. This CONSTRUCTED LINEAR NATURAL LAW OPERATOR is a LINEAR NATURAL LAW OPERATOR, and may be used in later calculations in the same ways as any other.

A1, A2, and A3 express long-accepted results used, explicitly or implicitly, since before Maxwell's time, and still used. But if A1-A3 are applied to models of coupled physical circumstances, without more, we come to terms that Maxwell called "difficult to interpret." The following frequently recurring fact causes a recurrent kind of problem.

Fact: If we construct terms of a finite increment model that correspond to N physical effects in interaction together, each per unit length (area, volume, time), we get a term which has N length (area, volume, time) increments associated together.

Problem: Since the equation is set out to correspond to only ONE length (area, volume, time), what do we do with the N-1 other lengths (areas, volumes, times) ?

THE STANDARD ANSWER NOW IS TO APPLY A LIMITING ARGUMENT TO THESE

N-1 LENGTHS (AREAS, VOLUMES), which invariably sets the crossterms to 0. When this is done, a procedure from the domain of the algebra is applied to another domain. A limiting argument is applied to physically senseless finite increment terms of the form

compound linear natural law operator

times

increment expression that refers to nothing identifiable

times

increment expression that refers to the model geometry

times

independent variable.

Such a finite increment term is geometrically undefined, dimensionally inconsistent with other terms in the equation in which it occurs, and arithmetically inconsistent from one system of units to another.

Appendix 1   shows, in detail, the ill definedness, dimensional inconsistency, and arithemetically inconsistency with unit system change that these conventional terms have.

KEY ASSUMPTION: We can avoid these difficulties and make sense of these terms if we assume that the extra increments can be, and must be

interpreted as dimensions (length, area, volume, time)

or

interpeted as dimensions times the unity operator

[ (1 unit length); (1 unit area); 1 (unit volume) ]

or

interpreted in an algebraic simplification of a compound natural law operator including increments, with every natural law operator implictly converted to a scale where the increment scale is unit scale. After algebraic simplification, the compound natural law operator formed is reconverted to the unit system being used.

These three forms of the KEY ASSUMPTION are operationally the same.

If we make this KEY ASSUMPTION, then A4 permits us to represent all the terms that occur in our physical modeling in a way that matches the physical model they correspond to geometrically and dimensionally. The terms formed are arithmetically consistent from one unit system to another.

A4) Axiom: CONSTRUCTED NATURAL LAW OPERATORS INCLUDING INCREMENTS: Algebraic formulation

Suppose a term is written as a result of a valid algebraic model construction, and two or more natural law operators and one or more spatial increments occur in the term. Let the axiom of geometrical consistency require that these natural law operators and one or more of these spatial increments be grouped together. THEN the natural law operators and spatial increments that are grouped together can be algebraically simplified together, multiplying (dividing) numerical parts and adding (subtracting) unit exponents to form a CONSTRUCTED LINEAR NATURAL LAW OPERATOR. This CONSTRUCTED LINEAR NATURAL LAW OPERATOR is a LINEAR NATURAL LAW OPERATOR, and may be used in later calculations in the same ways as any other.

A4 corresponds to a PHYSICAL implication:

A5) Axiom: CONSTRUCTED NATURAL LAW OPERATORS INCLUDING INCREMENTS: Physical formulation

There can be PHYSICAL interactions between several kinds of physical laws, that occur over a length, or over an area, or over a volume, or over a period of time. These effects may be expressed by CONSTRUCTED NATURAL LAW OPERATORS (DIMENSIONAL PARAMETERS) that include both NATURAL LAW OPERATORS and spatial increments.

According to A5), and axioms A1-A4), some terms now called infinitesimal, and some terms now called infinite, are finite terms. We can show how this comes about mechanically, on the basis of the assumptions A1-A4. We can then consider the results, in terms of cartographic reasonableness, and in comparison to data.

After a construction in the concrete domain of the measureable is done, mapping from this more detailed map to a simpler, more abstract domain (one of Hart's Typed Family of Fields, commonly thought of as the domain of the algebra, is straightforward. It may be done by taking a limit of the model equation. For differential equations, this is the same as transfering the equation, symbol for symbol, into the more abstract domain.

Let's apply these axioms and derive a neural conduction equation, starting within a concrete domain of the measurable that corresponds to the physical conduction line that we are modeling.

To derive a differential equation from a physical model in classical physics (still the dominant physics in practical work) we argue as follows.

1. We construct a model (including a sketch, and any necessary information) at a finite scale that represents the physical laws and geometry in question.

2. We derive finite increment representative equation(s) in a concrete measurable domain. We set up a representation that maps the model completely (or, if the equations involve some recursive relation, we represent the model to some specified degree of completeness.)

3. We define each term in each equation so that it fits our model geometrically and dimensionally.

4. We then map our equations, fully defined in 3, into the domain of the algebra.

Let's proceed to infer a finite increment equation for electrical transmission along a line, a case that is inherently coupled. Fig. 1 shows a neural conductor (axon or dendrite considered as a transmission line). A tubular membrane is filled with and surrounded by an ionic fluid. The fluid inside the tube carries current (and signal) and has resistance R and electromagnetic inductance L per unit length. The outer fluid is grounded. The membrane separating these conducting fluids has capacitance and leakage conductance per unit area. We speak of the following variables and parameters:

v = voltage                      i = current
x = position along the line                alpha= arbitrary length interval
R = resistance/length                          L=electromagnetic inductance/length
G= membrane conductance/length                   C=capacitance/length

Fig. 1 shows an arbitrarily chosen length , which we will call x because that is commonly done. is picked from other indistinguishable lengths. (For consistency, the length of is a number times the unit of measure used in the calculation in the x direction.) We are giving length a two-symbol name that includes a separate number, . Other entities that have numerical values in our calculations are denoted as single symbols, and do not have separate numerical symbols (such as ) associated with them.

We need finite difference equations, v(x,x,t) and i(x,x,t). We face the following coupled circumstance.

Because voltage drop over a length depends on current, and current over length partly depends on charge carriers stored in capacitance or lost through membrane leakage over , we know that v/x for x= must partly depend on effects including C and G over length x

and

Because current drop over length depends on voltage change over , and voltage change over partly depends on R and L, we know that i/x for x= must partly depend on effects of R and L over length x.

From such interactions, it follows that:

i over the interval is a function of v at x and x+x

which is a function of i at x and x+x

which is a function of v at x and x+x

which is a function of i at x and x+x

and so on

The logic for v is symmetric.

v over the interval is a function of i at x and x+x

which is a function of v at x and x+x

which is a function of i at x and x+x

which is a function of v at x and x+x

and so on

Our i equation implicitly contains our v equation (and its derivatives.) Our v equation implicitly contains our i equation (and its derivatives.) When we write out our v and i equations more explicitly, we still have implicit couplings between i and v functions. In turn, we can write these implicit equations out more explicitly. We still have couplings. As we write our relations out in more and more explicit form, we get more and more terms, in an endless regress.

An expansion according to this logic is set out in Appendix 1. The expressions are set out in a CONCRETE DOMAIN OF THE MEASURABLE according to algebra, without yet interpreting the terms geometrically and dimensionally.   Our crossterms in () are not yet clearly related to the model that generated them.

1. For a single term, we could make different interpretations of the symbols in the term that apply to different geometries.

2. For a single term, we could make different interpretations of the symbols that would correspond to different units.

Consider one term, grouped in different algebraically permitted ways These different groupings, that are algebraically the same, have different physical interpretations.   In 7b we interpret every crossterm of 7a in this same way, so that every crossterm consists of an independent variable, an interval x, and a compound linear natural law operator (dimensional parameter) shown prior to algebraic simplification.   the compound natural law operators (dimensional parameters) simplify to   In 7c, the natural law operators and constructed natural law operators are broken down into their numerical and dimensional parts. Every natural law operator and constructed natural law operator has a dimensionality of vn tm Qo xp with dimensionality expressed in ratio form (negative dimensionality in the denominator, positive in the numerator.) Dimensions to the 0 power are not shown. Every term has net dimensions (of voltage) as it must.

So far, all we have done is INTERPRET what the terms in 7a MUST mean to satisfy dimensional consistency and geometrical consistency in the CONCRETE DOMAIN OF MEASUREMENT in which the conduction model from which they were derived exists.

To map the model into the domain of the algebra, we must take a limit. Here is the basic logic to used to map from the concrete domain into a differential equation in the domain of the algebra.    Our abstract equations may be written more simply, by making our dimensional notation implicit.   This differential equation, in the domain of the algebra, is subject to exactly the same rules as other differential equations.

However, it contains terms, corresponding to the crosseffects in 7a, that would have been called "infinitesimal" according to the current practice, which applies a limiting procedure to geometrically and dimensionally undefined terms. Whether these finite terms are indistinguishable from 0, or very large, depends on the numerical size of the natural law operators (dimensional parameters) R, L, G, and C.

Comparison of a wire case and a neural dendrite case shows the contrast in values of these crossterms that can occur.

For a 1 mm copper wire with ordinary insulation and placement, typical values of the natural law operators (dimensional parameters) would be:

R = .14 x 10-4 ohm/cm             C = 3.14 x 10-9 farads/cm

G = 3.14 x 10-10 mho/cm                L = 5 x 10-9 henries/cm

Here is (1) with the numerical value of terms set out below the symbolic expression of the terms: Here are the corresponding natural law operator (dimensional parameter) values for a 1 micron diameter neural dendrite, assuming accepted values of axolemma conductivity, capacitance per membrane area, and inductance per unit length.

R = 1.4 x 1010 ohm/cm L = 5 x 10-9 henries/cm

C = 3.14 x 10-10 farads/cm G = 3.14 x 10-8 mho/cm

This R is 1011 larger than in the previous case of the wire. The two cases modeled by the same equation are radically different. Cross product terms that are trivial in the wire case are dominant in the neural dendrite case. In the wire case, calling these crossterms "infinitesimal" is a near-perfect approximation. In the dendrite case, calling these crossterms "infinitesimal" is gross error.

I have shown a different approach to deriving differential equations from physical models, based on different assumptions and procedures. We may ask three kinds of questions about these changed derivations.

First, we may provisionally grant the assumptions, and then ask how

the derivational consequences work out.

The derivation of the S-K transmission equation shows those consequences. Under the new procedure, when there are coupled circumstances, coupled terms now called infinitessimal will be finite. The size of these finite terms will depend on the size of the natural law operators in the model under consideration.

Secondly, we may ask if the assumptions of the new procedure are required for symbolic consistency in the derivations. A major test is consistency from one unit system to another, an issue that applies to measurable circumstances, but that is not a concern in the abstract domain of the algebra.

The loop test shows this.

Thirdly, we may ask if the new assumptions yield derivations that

FIT DATA that is not consistent with the standard derivations.

I have a linear algebra derivation of the conduction equation
that fits. Spice simulation of the transmission equation

Neural data fits.

Soon, I hope, QM data will fit.

Appendix of this Draft

Some terms in finite increment equations that are derived from abstract algebra without further interpretation do not refer to model geometry, are dimensionally inconsistent with the equation in which they occur, and are arithmetically inconsistent from one unit system to another. Now, these terms are discarded as infinitessimal without considering these definitional difficulties. The mapping into the concrete domain of the measurable avoids these difficulties according to procedures that make these terms finite. Let's take a closer look at what these logical difficulties are. Let's consider the terms in equation 7a that are conventionally dismissed by a limiting argument.

These terms were all constructed to refer to a particular model with = x.    On the right hand side of the first line there are two terms, that are not dismissed by limiting arguments, that refer clearly to model geometry. These terms ( ) are dimensionally consistent with each other and with the left hand side of the equation. These terms are arithmetically consistent from one unit system to another In the terms in (), an increment, x, refers directly to length in Fig **. A natural law operator (R or L) defined per unit length, acts over length x, and transforms a variable (i or di/dt) numerically and dimensionally. The dimensionality of these terms is voltage, with all the other dimensions taken to the exponent 0, as it must be.

These terms pass a conservation test that representations of physical circumstance must pass. If x is divided into N pieces, and the expressions evaluated over those N subintervals are computed and summed, that sum is the same as it would be for x taken in one step.

These terms are also arithmetically consistent from one unit system to another. Let's look at a simple loop test, analogous to many closure tests in physical logic. In Fig. ~1, an algebraically unsimplified term such as one of those in ( ) is set out in cm length units at A. This quantity is algebraically simplified directly in cm units to produce "expression 1." The same term is also translated from A into a "per meter" basis at C. The translated quantity at C is then algebraically simplified to D in the same conventional fashion. The expression at D, expressed in meter length units, is converted to a "per cm" basis to produce "expression 2." Expression 1 and Expression 2 must be the same, or the calculation is not consistent with itself. Quantities like those in ( ) pass this essential loop test. These terms are arithmetically consistent from one unit system to another. Below the first line of ( ) are terms in finite increment equation ( ) that are typically dismissed by a limiting argument. At finite scale, these terms fail the tests the terms in ( ) pass. These terms do not refer to the model geometry that they were constructed to represent. These terms are dimensionally inconsistent with the equation in which they occur. These terms

are arithmetically inconsistent from one unit system to another.  In the terms in ( ), there are one or more increments, labelled x, that cannot refer directly to length in Fig **. It is not clear how the x symbols in these terms refer to the physical model these terms were constructed to represent.

These () terms are dimensionally inconsistent with the equation in which they occur. To see this, it suffices to see that these terms all fail a conservation test that representations of physical circumstance must pass, and that the terms in () do pass. For the terms in (), if x is divided into N pieces, and the expressions evaluated over those N subintervals are computed and summed, that sum is NOT the same as it would be for x taken in one step. The whole does not equal the sum of its parts. With conventional patterns of interpretation, we cannot say that these terms are equal to a compound dimensional parameters expressed "per unit length" times a length increment corresponding to model geometry. These terms are somehow "per unit length2", of "per unit length3," and are therefore dimensionally inconsistent with the terms in ( ), which are "per unit length."

These ( ) terms are also numerically meaningless at finite scale, as shown below. A "number" or "expression" that can be manipulated by "proper arithmetic" and permissible unit changes so that it has any value at all is meaningless. Let's look at the same loop test referred to before in Fig. 666. An algebraically unsimplified term in () is set out in cm length units at A. This quantity is algebraically simplified directly in cm units to produce "expression 1," dealing with the dimensional parameters and increments as "just numbers." The same physical quantity is also translated from A into a "per meter" basis at C. The translated quantity at C is then algebraically simplified to D in the same conventional fashion. The expression at D, expressed in meter length units, is converted to a "per cm" basis to produce "expression 2." Expression 1 and Expression 2 must be the same, or the calculation is not consistent with itself. Terms like those in the curly brackets consistently fail the loop test.   The loop test fails!

The loop test fails because a standard procedure is flawed when applied to terms like those in ().

Before algebraic simplification, going from one unit system to another adjusts not just the numerical value of dimensional properties in the different unit systems, but numerical values corresponding to the spatial variable (length), as well.

After algebraic simplification, one has a compound dimensional property - adjusting it to a new unit system corresponds to adjusting numerical values that correspond to the unit change for the dimensional properties only, with no corresponding adjustment for the absorbed spatial variable.

The result is an irreversible, numerically absurd, but now standard mathematical operation. The difficulty has been deeply buried and hidden in our notation.

Note that the loop test of Fig. 666 only fails for terms that are now routinely discarded as "infinitesimal" or "infinite" without detailed examination.

Derivation of equations such as () according to the rules of a concrete domain of the measurable, as described here, avoids these logical difficulties. The occurrence of these logical difficulties should cast doubt on the notion that derivations such as the derivation of the Kelvin-Rall transmission equation are "rigorous" and scientifically true derivations.

1. Reality Bytes, NYT forum

2. Johnson, G. (1997) Proteins Outthink Computers in Giving Shape to Life NEW SCIENTIST March 27, 1997.

3. foot definition

4.