

The theory of catastrophes and homeostatics uses a mathematical model to transfer the hereditary information in an invention problem.^{2,3,4} Transfer of the hereditary information is the transfer of physical properties from the technical contradiction to the new solution (or Xelement). There is a biological analogy: opposite properties of the technical contradiction are "parents" and the Xelement is their "baby."
A search of physical properties of an Xelement is the important problem in ARIZ. If the Xelement is a field, the problem solver needs to identify that X. For example, is it a field of pressure or an electric field? If the Xelement is a substance, the solver wants to know its physical properties. Physical properties are designated by letters (x,y,z) in the hereditary mathematical model. This article represents an attempt to transition from mathematics to physics. Decoding Bartini's method is one such way.
Roberto Oros di Bartini (18971974) was a Soviet aircraft designer and inventor who outstripped the constructive ideas of his contemporaries to no less than 50 years.^{5} This charismatic and mysterious personality can be compared to the scale of the person and merits of Nikolo Tesla. He graduated from Milan Polytechnical Institute in 1922. After the Fascist revolution in Italy in 1923, he worked in the USSR as an aviation engineer and designed 60 different planes.
Bartini developed engineering creativity too. The "andand" method by Bartini is more than 20 years older than TRIZ. Both methods have grown from dialectic logic independently and at various times in despite of some analogy. There is an opinion that his methods developed from mathematics.^{6} Bartini wrote two theoretical physical articles in which he used a dimensional method for physical sizes. By connecting a dimensional method with a hereditary mathematical model, Bartini's method has been deciphered.
The ARIZ model contains a conflict pair, i.e., the tool and an object. The tool has a useful and harmful action on an object. When the tool is in one condition, the first action is useful, but the second action is harmful. Under opposite conditions, the first action becomes harmful, but the second action becomes useful. There is a need to find an Xelement to eliminate the harmful action while not disturbing the useful action.
The mathematical model of the conflict pair and Xelement is set by cusp catastrophe with potential function:
E(x) = 0.25 x^{4} – 0.5 l x^{2} – mx (1)
where x = coordinate of state of object, l = tool control parameter, m = Xelement control parameter
The minimum of potential function E(x) is the purpose of invention solving, therefore the size of potential function is equal to the undesirable effect. Let's consider the dimensional analysis E(x) by means of examples.
A capillary needs to seal after filling an ampule with fluid. The flame of a gas burner seals the ampule capillary, as shown in Figure 1. The flame blazes irregularly, so there are sometimes defects. Technical contradiction: If a flame is greater, the ampule is sealed well, but the fluid is overheated; if a flame is small, the ampule is sealed poorly, but the fluid is not overheated. The flame is a tool and the ampule is an object. The location of the ampules in water and extremal reinforcement of flame is the contradiction to be resolved.

In this case the height x of an ampoule from the top of a capillary defines zones of undesirable effect, shown in Figure 2. The height will be coordinate x of an ampule. By applying formula 1, every item should have the physical dimension, x^{4}, i.e., length in the fourth degree (L^{4}). Then we can find the physical dimension of control parameter l for the tool – it has the dimension of a surface (l = S =L^{2}).

This surface can be presented as an operative zone, i.e., place of contact for the tool and an object. Now we need to find the dimension of control parameter m for an Xelement. It has dimension of volume (m = V = LS =L^{3}) – the Xelement (water) is placed in volume in the control answer of this problem.
A pneumatic stream carries small steel balls in a pipe. The pipe has a knee for the turn of a stream, as shown in Figure 3. The pipe wears out at turns because of the impacts of the balls.

The pipe with a knee is the tool and the balls are the object. There are two states of the tool: 1) when a knee has a big radius (R>) and 2) when a knee has zero radius (R<). The fast turn is the part of the main manufacturing process and we choose zero radius (Figure 3, center). At once we find an operative zone – the internal surface of a pipe at a turn. The operative zone has twodimensions, defined by control parameter l of the tool under formula 1. Hence, the volume is control parameter m of an Xelement. Then the Xelement is a volumetric spatial resource, i.e., a pocket. The balls fill a pocket and eliminate the pipe's deterioration. At last it is possible to believe that the state of an article is defined by a contact line of the ball and pipe.
When we split a tree trunk, the wedge jams in wood. The wedge is the tool and the tree trunk is the object. If the width of splitting (x = L) is a state of the object, then the operative zone should be twodimensional. The lateral surface (L^{2}) of a wedge is twodimensional. The solution is a double wedge with one part made of lowmeltingpoint material. The internal space of a wedge is volume (L^{3}) and control parameter of an Xelement as shown in Figure 4.

The solution is a double wedge with one part made of lowmeltingpoint material. The internal space of a wedge is volume (L^{3}) and control parameter of an Xelement as shown in Figure 4.
These three examples lead to the following conclusions:
Do other trends exist? Yes – and this can be found with Bartini's LT system of units.^{9,10}
Bartini's ideas have appeared in TRIZ.^{5,6,11,12,13} The author suggests using Bartini's system of units together with TRIZ on a new mathematical basis.
In 1873 J.C. Maxwell offered a system for measuring using two sizes: length (L) and time (T). Bartini established relationships between physical constants and also offered a kinematic system of physical values.^{9} Next he and Pobisk Kuznetsov developed a geometrical direction in research of physical dimensions.^{10} Bartini's system resulted in the LTtable (Table 1).
Table 1: LTTable – Kinematic System of Physical Values (Length [L] vs. Time [T])  
Values  L^{2}  L^{1}  L^{0}  L^{1}  L^{2}  L^{3}  L^{4}  L^{5} 
T^{6}  L^{2}T^{6}  L^{3}T^{6}  L^{4}T^{6}  L^{5}T^{6}  
T^{5}  L^{1}T^{5}  L^{2}T^{5}  • Surface power  L^{4}T^{5}  Power  
T^{4}  L^{0}T^{4}  • Specific gravity • Gradient of pressure  • Pressure • Tension  • Surface tension • Rigidity  Force  • Force momentum • Energy • Statistical temperature  
T^{3}  L^{0}T^{3}  Current density  • Electromagnetic field strength • Ductility  Current • Loss mass  • Motion quantity • Impulse  • Angular momentum  
T^{2}  nbsp;  • Mass density • Angular acceleration  • Magnetic displacement • Acceleration  Potential difference  • Mass • Quantity of magnetism or electricity  Magnetic momentum  Moment of inertia  
T^{1}  Volume charge density  Frequency  Velocity  • Twodimensional abundance • Velocity of change of the area  Loss volume  L^{4}T^{1}  L^{5}T^{1}  
T^{0}  L^{2}T^{0}  • Crookedness • Change of conductivity  Dimensionless constants (for example, a radian)  • Length • Capacity • Selfinduction  Surface  Volume of space  Distribution of volume along length  
T^{1}  Changing of magnetic permeability  Conductivity  Period • Duration  Time necessary for change of length on unit  L^{2}T^{1}  L^{3}T^{1}  
T^{2}  Magnetic permeability  L^{1}T^{2}  Surface of time  L^{1}T^{2}  L^{2}T^{2}  
T^{3}  L^{2}T^{3}  L^{1}T^{3}  Volume of time  L^{1}T^{3} 
The infinite vertical columns of the kinematic system contain a series of integer degrees of length (from L^{–2} to L^{5}) and infinite horizontal rows contain integer degrees of time (from T^{–6} to T^{3}). The crossing of each column and every row gives the dimension of a certain physical quantity. Dimensions of all physical values are represented as a product of integer degrees, L^{n}T^{m}, where n+m ≤ 3 for threedimensional space.
The LTtable expresses physical laws of conservation. Intuitively it is clear that the LTtable should have important practical value for inventorrelated work, but Bartini did not provide data on its application in technical creation.
Let's compare Bartini's LTtable to TRIZ tools. The contradiction matrix is the most similar object, containing 39 features for technical contradictions (for instance, length, volume, time, speed, pressure, force, temperature). More abstract features can be expressed by means of other measured physical values. For example, the shape can be estimated by aerodynamic resistance, and the maintainability can be estimated by average time of restoration after failure. These 39 features name vertical columns and horizontal rows in the contradiction matrix. Similar features are placed in the cells of the LTtable. When the technical contradiction (TC) is formulated, the problem solver searches for the cell from the crossing of bad and good features of the TC. This cell contains numbers of the principles of invention – the numbers are an output of the problem solving process. (See Figure 5.)

Consider that the technical contradiction (or system model) is an input in the LTtable also. The technical contradiction contains the object, the tool, the good feature and the bad feature, depending on the states of the tool. The object is set by the coordinate of a state X and the tool is set by the control parameter m.
In Table 1, look at horizontal row T^{0}. The fragment L^{0}T^{0}, L^{1}T^{0}, L^{2}T^{0}, L^{3}T^{0} is known as TRIZ sequence "pointlinesurfacevolume" (the contact form between the tool and object), but the fragment L^{1}T^{0}, L^{2}T^{0}, L^{3}T^{0}, L^{4}T^{0} is sequence "objecttoolXelementuseful effect" for examples 13. It is supposed that the useful effect and undesirable effect E(x) have identical dimensions L^{4}T^{0}=L^{4}. The difference consists only in a sign (+ or ). Then the cell L^{4}T^{0} shows the distribution of the Xelement volume of the length of an object. There should be a certain distribution of water volume along the height of an ampule (example 1), the certain distribution of a pocket along a contact ballpipe line (example 2) and the certain distribution of lowmeltingpoint metal on width of split (example 3).
Cells L^{1}T^{0} and L^{2}T^{0} are an input of the LTtable and cells L^{3}T^{0} and L^{4}T^{0} are an output of the LTtable. The cell L^{3}T^{0} defines a spatial resource of the Xelement, i.e., volume. The cell L^{4}T^{0} sets the state of the certain accommodation of this volume.
Consider the converting mechanism from an input to an output. The Xelement dimension is equal to the product of object dimension and tool dimension, i.e., (L^{1}T^{0})∙(L^{2}T^{0}) = L^{3}T^{0} = L^{3}. The dimension of the desirable effect is equal to the product of the Хelement dimension and the object dimension, i.e., (L^{3}T^{0}) * (L^{1}T^{0}) = L^{4}T^{0} = L^{4}. The general trend of development of a spatial resource is shown in Figure 6.

Other horizontal rows of the LTtable allow for the construction of trends of the spatial development of resources. Each cell of a temporal trend differs from the next cell of the same trend on a multiplier T. Diagonal lines of the LTtable are the Sufield trends of resources. Each cell of a Sufield trend differs from the next cell of the same trend on a multiplier L^{1}T^{1}, i.e., velocity V = L^{1}T^{1}.

Lines of development (or trends) form triangles of resources. Cathetuses of the triangle are trends of spatial and temporal resources and the hypotenuse is a trend of Sufield resources. It is possible to assume that the resource triangle will allow a transformation of spatial and temporal resources in the resources of substances and fields. Return transformation is possible too if a Sufield trend is decomposes on the orthogonal axes of the LTtable.
Consider the resource triangle ABC (Figure 7, red outline). The horizontal cathetus AB of triangle ABC is a fragment of a trend of spatial resources with dimensional trend L^{0}T^{0}, L^{1}T^{0}, L^{2}T^{0}, L^{3}T^{0}. The elements of cathetus "pointlinesurfacevolume" are essences. Figure 8 shows that the new essence is the sum of at least the two previous essences: point and point = line, line and line = surface, surface and surface = volume. The line, therefore, receives some property of a point in the inheritance, the surface receives some property of a line in the inheritance, the volume receives some property of a surface in the inheritance and so on. There is a general property that passes from a point to volume by right of succession. This general property refers to as assemblage of points or population of points. The point is an assemblage of points and the line is and assemblage of points and the surface is assemblage of points and so on. The general property "runs" along a trend – the transfer of the hereditary information. The elements of a cathetus are generations.

The general property of a trend is set by dimension. Dimension L^{n}T^{0} of the new essence is product (L^{n1}T^{0})∙(L^{1}T^{0}) where L^{n1}T^{0} is the dimension of the previous essence. The multiplier L^{1}T^{0} enters into dimensions of all elements of a spatial trend, defining the general property of a spatial trend in the mathematical image. Multiplier L^{1}T^{0} is a gene of a spatial trend or genetic length, too, as L^{1}T^{0 }= L^{1}. The dimension of generation is equal to the product of the dimension of the previous generation and genetic length as seen in Figure 9a, which represents the mathematical model of the process of transfer of the hereditary information. Blocks of multiplication are elements of this model and dimensions are input and output signals of blocks.
a = structured scheme, b = equivalent electrical circuit 
The genetic length is a synlengthing signal (similar to a synchronizing signal). The output signal of the block exists when the input signal exists and the synlengthing signal exists, i.e., output signal = input signal and synlengthing signal. The sign "&" designates a logic multiplication or conjunction (andand). The hereditary information is transferred from point A to point B if each generation of cathetus AB exists. The equivalent electric circuit is shown in Figure 9b, where switches S1,S2,S3,S4 simulate the operation of logic multiplication. Hereditary information passes from A to B when S1 and S2 and S3 and S4 are closed.
The temporal resource trend is defined similarly. The gene of time is equal to L^{0}T^{1} and genetic time is equal T^{1}. Consider movement along cathetus BC. Point B has dimension L^{3}T^{0} and the essence is volume. The next cell has dimension (L^{3}T^{0})∙(L0T1) = L^{3}T^{1} and the essence is the velocity of change of volume. The next essence is an acceleration of change of volume. Point C is the velocity of change of acceleration of change of volume.

Trends of Sufield resources are represented by the color diagonals for n+m ≤ 3 seen in Figure 10. All trends of Sufield resources transfer a gene of velocity L^{1}T^{1} from generation to generation. The gene of velocity, therefore, is an invariant of the totality of Sufield trends. A distinction of Sufield trends consists in the sum S_{n+m} = n+m of exponents n and m for dimensions L^{n}T^{m}. The yellow trend has the sum S_{n+m} = 0 and gene L^{n}T^{n}. The grey trends have sums S_{n+m} = ±1 and gene L^{n}T^{n±1}. Blue trends have sums S_{n+m} = ±2 and gene L^{n}T^{n±2}. Green trends have sums S_{n+m} = ±3 and gene L^{n}T^{n±3}.
A socialbiological analogy can be used here. The trend forms a family of several generations. The totality of trends (or families) forms mankind. Each essence of the LTtable has a trend attribute S_{n+m}, the same way each person has a family attribute. Each essence has the same attribute L^{1}T^{1} of trend totality the same way each element of mankind has an attribute of a person.
The hypotenuse AC of a resource triangle is a fragment of a yellow trend of Sufield resources. The dimensional length of cathetus AB is AB = L^{3}T^{0} / L^{0}T^{0} = L3. The dimensional length of cathetus BC is BC = L^{3}T^{3} / L^{3}T^{0} = T^{3}. The dimensional length of hypotenuse AC is AC = L^{3}T^{3} / L^{0}T^{0} = L^{3}T^{3} = V^{3}. Then the Pythagorean theorem applies to a resource triangle: (AC)^{2} = (AB)^{2} and (BC)^{2} = (L^{3})^{2} and (T^{3})^{2} * (L^{3}T^{3})^{2}, where "and" is logic multiplication.
Bartini used a method of logic multiplication "andand" in his work as an inventor. Logic multiplication summarizes essences by multiplying their dimensions. The addition of essences is a necessary condition for the search of physical properties of an Xelement. The following system of the differential equations in works is [2, 4]:
Kdx / dt = 3xy + az (1)
Kdy / dt = 3xy  az (2)
Kdz / dt = 3xy  az (3)where x and y = the coordinates describing a change of competing properties of the technical contradiction, z = the coordinate describing a change of property of an Xelement, K and a = factors
The equations (13) describe a process of thinking after insight. Property z of an Xelement is defined completely in the established mode, i.e., 0 = 3xy – az and
z = q * xy, q = 3/a (4)
Equation (4) is fair for dimensions, x = L^{n1}T^{m1}, y = L^{n2}T^{m2}, z = L^{n3}T^{m3}. If a new essence with hereditary attributes needs to combine with two old essences, we should multiply their dimensions. Though the dimensions are multiplied, the hereditary attributes of different trends (exponents n, m) are summarized. A "baby" L^{n3}T^{m3} comes to a new trend of Sufield resources still having the hereditary attributes of its parents. L^{n3}T^{m3} is the carrier of the hereditary information of a new trend to all other elements.
There is a second type of input in the LTtable – useful and harmful properties of the technical contradiction are entrance parameters. The resource of the Xelement will be an output of the LTtable. If useful and harmful properties of the technical contradiction are established, suitable physical dimension of these properties should be found in LTbasis. Then the dimensions are multiplied. The product gives a cell of a new Sufield trend in the LTtable. Suitable physical properties of the Xelement are in diagonal cells of a new trend. There is also the opportunity to use the names of columns and rows of the contradiction matrix for entrance parameters.
Consider a continuation of example 1 (soldering ampules). The initial technical contradiction is: If the flame is greater, the ampule is sealed well, but the fluid spoils; if the flame is small, the ampule is sealed poorly, but the fluid does not spoil. How is it possible to measure good or bad soldering of an ampule and how is it possible to measure damage of the fluid? Assume that the length of the soldered capillary defines the quality of soldering and temperature defines the quality of the fluid. The Xelement, therefore, should have hereditary attributes of the temperature of the fluid and the big length of the soldered capillary. "And" the small temperature "and" the greater length provide a positive effect. The dimension of temperature is L^{5}T^{4} and the dimension of length is L^{1}T^{0}. (See Figure 11) Multiplication results with (L^{5}T^{4}) * (L^{1}T^{0}) = L^{6}T^{4}, leading to a trend with a hereditary attribute of L^{n}T^{n+2}. The movement on this trend (along a red arrow) gives a suitable Sufield resource for the Xelement, i.e., loss of volume with the dimension L^{3}T^{1} = m^{3} / c. It is known that the volumetric discharge of a hot or cold carrier has such a dimension in thermal systems. Flowing water can be a cheap resource.
The first part of an example 1 gives a spatial resource of an Xelement as volume. Continuation gives a Sufield resource as the volumetric discharge (volume/time).
A steel spring has the following limitations:
It is necessary to find a simple way to increase the rigidity of a spring. This problem has only one physical value – the rigidity of a spring. Rigidity has the dimension L^{3}T^{4} in the LTtable. The second physical value cannot be entered for formation of the technical contradiction. It must be formally entered – it has zero dimension L^{0}T^{0}. Multiplying leads to (L^{3}T^{4}) * (L^{0}T^{0}) = L^{3}T^{4} – remaining in the same trend. Movement on this trend (along a lilac arrow) gives a Sufield resource for the Xelement, i.e., electromagnetic field strength with dimension L^{2}T^{3}. TRIZ suggests magnetizing coils of a spring, then there is an additional repellent strength at compression of a spring.

Figure 12 shows a drawing of a floating tiltmeter – when the base tilts, the float emerges and turns a rod around an axis of the hinge. The indicator (not shown) is on an axis of the hinge. Attempt to increase the sensitivity of the tiltmeter by means of an increase in diameter of a float. A large volume of float leads to smaller errors in measurements. Hydrodynamical resistance, however, increases with variable inclinations and dynamic mistakes increase.

Now use Bartini's principle: "and" the volume of a float should be greater "and" hydrodynamical resistance should be small. The volume has the dimension L^{3}T^{0}. Hydrodynamical resistance has the same dimension same as force L^{4}T^{4}. Multiplying these results in (L^{3}T^{0}) * (L^{4}T^{4}) = L^{7}T^{4} – to a trend with hereditary attribute L^{n}T^{n+3}. Movement on this trend (seen along a red arrow in Figure 13) provides a suitable Sufield resource for the Xelement, i.e., moment of inertia. The float is a mobile element and has inertia. Its form (inertia) can be changed – for example, make a float in the form of a wing.

Dimensional LTanalysis is wide and can serve as the physical and mathematical body for TRIZ.
The author thanks the Russian Fund of Federal Property for support of work on grant № 060801289a.
Alexandr B. Bushuev is assistant professor information technologies, mechanics and optics at St. Petersburg State University of Russia. Mr. Busheuv has been teaching TRIZ for more than 20 years. His scientific interests include the mathematical bases of the TRIZ, automatic control, methods of optimization, economic cybernetics and patent expert examination. Contact Alexandr B. Bushuev at bushuev (at) inbox.ru.