At various points in the Foreground and Background pages of this website key points arise from “models” and “simulations”. What do we mean by these terms?
Essentially, models are constructed to explore aspects of the physical world that are not directly measureable or accessible by the senses. In fact, one might even consider that normally such simulations may be employed to interpret what the senses might actually sense – but more on this later.
A model may conveniently be considered as a “Black Box”, with inputs that are directly measureable or sensed. This Black Box may have parameters that control its inner workings and its outputs provide new information. Of course the input information may be nonsensical or the model operation erroneous, making any new information arbitrary and useless. Outputs need to be useful for the model to be valuable. Take a television, for example, where the input radio signal is received through an aerial, the inner circuits are tuned to receive a particular frequency, the numerical content is decoded for the channel (parameter) that is selected and the output is displayed to inform and entertain the viewer. It is not necessary to understand exactly how a television works to appreciate its value.
Innovation itself may be treated using a Black Box approach, by converting empirical observations and scientific research into information that has utility and value. In this application we have connected the models that make up the tools of scientific research to the mysterious Black Box through which value created through the researcher’s endeavour is appreciated.
For further information see: When Science Meets Innovation: a new model of research translation
We have modelled the subject of value appreciation using a Value Surface that maps perception of value across a statistical population of consumers. The elevation of this Value Surface in a hypothetical economic potential field provides a means to link investment and the innovative endeavours of an enterprise to value created. Let us consider why a model might be useful to explore such a potential field.
A Flat-Earth Person Finds Dimension Three
Imagine that you are a flat-earth person. Not one of a conventional, three-dimensional kind who is able to believe that at some point you may slide off the edge of our saucer-shaped planet. Rather imagine that the force of gravity in some way acts to compress your perception of height to an infinitesimal thinness. You will still live on Earth as you do today, but how different your view of the planet would be. You would truly be a two-dimensional person. Let us consider what you may see and how you will fit these observations into an understanding of your world.
If you are initially resident on a horizontal plane, then this surface will stretch out before you. At various points in the distance the altitude of the terrain may change. Any increase in height, even if this is just a gentle slope, will be seen as an impenetrable barrier. Likewise, a real lower piece of ground will appear as a hole to oblivion. You will see the edges of these barriers and holes as lines of constant height, just as contours appear on an Ordinance Survey map.
Your task today is to move to your next appointment, for which you have a map giving the details of your journey. This map may be a length of string containing paired instructions of distance and direction. First go 0.7km at 123 degrees, which should bring you to a hill. Your perception of this hill as an insurmountable barrier does not change but something strange happens as you begin to climb. The plain on which you approached instantly disappears from view. Facing you now is a solid vertical wall, behind is a limitless abyss. You only perceive the linear contour that wraps around either side of the hill on which you climb and disappears from view. But something is happening as you move forward. This contour is constantly changing and, more importantly, you are using energy although this does not seem to be having any effect. You are not afraid as the steps you are taking and their associated changes of contour are all precisely detailed on the map you have, so that you cannot possibly be lost!
Finally and suddenly, the hill you have climbed breaks into a plain and again your full two-dimensions of perception are restored. In the distance the new plain stretches away to further barriers and holes. Following your map, a descent into a hole is the reverse of the hill you have just climbed. You step into the abyss whilst behind, a perceived contour gives shape to an otherwise impenetrable barrier. But caution must be exercised here. Some holes, the really steep ones, should be descended with care and are best avoided. Energy is returned too fast to be easily dissipated by your bodily processes. And then there are the ‘strange’ holes. These are clearly identified on the string-map as areas to be avoided at all costs – few return from such a descent and, when they do, they are strangely wet.
Shadows lengthen as you move across the new plain. The impenetrable barriers radiate darkness and as the day moves into evening, this dark-radiation grows in intensity until the effects from all barriers superimpose. It is important for you to reach your destination before this complete darkness falls.
Before you fall asleep, satisfied with your two-dimensional endeavours of the day, you find time to unfurl another coil of string. It is a popular book on genetics and evolution and you learn from the string of characters how wonderfully optimised you are for two-dimensional survival by your genetic template. The mechanism is a model of natural selection through the duplication of a molecular double ziz-zag.
Your perception of the two-dimensional world is likely to be something akin to a large department store. It is a collection of planes containing interesting artefacts but separated by these impenetrable-appearing barriers. Up or down you must go to gain access to new vista. Rather like the opening of an elevator door.
But then someone, an innovator, imagines that gravity is acting as a potential field perpendicular to your perception. Such potential fields fill their space with a force that pulls or pushes onto things which enter their vicinity. Magnetism is one example. In this case, it is proposed that a constant downward force of attraction is acting as you move upwards through this field. The height you have gained is then proportional to the energy expended in the climb. This is a breakthrough. All two-dimensional surveyors now have to do is monitor the energy required to reach every point on the impenetrable barriers to calculate their height.
The method the two-dimensional cartographers used in charting their hidden territory was to move to a higher altitude and send back a signal to the starting point giving information on the height gained. Conservation of energy provides the basic principle. The tools of the cartographer are a ramp of fixed length L and a heavy cylinder of mass Mc. The ramp is used to reach up to a point of higher altitude as shown in the figure below. The cylinder is then released from the top and carries the signal revealing the height h to the bottom.
The cylinder and ramp of the two-dimensional cartographer
The ramp-cylinder system shown above is the cartographer’s Black Box within which the conservation of energy principle operates: –
Potential energy at the top = Kinetic energy of the rolling cylinder at the bottom
= Translational kinetic energy + Rotational kinetic energy
Mc.g.h = ½ . Mc.Vc2 + ¼ . Mc .Vc2
= ¾ . Mc .Vc2
So that h = ¾ . (Vc2 / g )
Here Vc is the velocity of the cylinder at the bottom of the ramp. Unfortunately, this velocity is not easy for our two-dimensional cartographers to measure, as the cylinder appears from nowhere to clatter off the end of the ramp. Some further analysis is needed and this is provided by the mathematical tools of calculus, which convert the measurement into one of time t from the moment the cylinder is released to the point it reaches the base of the ramp. In this case: –
h = (3.L2)/(g.t2)
and Sin (a) = h / L
All the mapmakers now need to do is to time the arrival of the cylinder and apply the above equation to know the height at which it was released. They may then move onto another point and repeat the process, to gain a full knowledge of the height and the slope a of their surrounding terrain.
The conversion of potential energy into the kinetic energy of the descending cylinder is used to create the information content of the two-dimensional map. The Black Box that provides the two-dimensional cartographers access to their third dimension contains a model that converts measurements that they can make, time in this case, into something meaningful that they cannot measure directly but which they need to know.
In principle models can be valuable, but only when they provide useful insights into the real world. And their value, like that of any product, will diminish as they become superseded by alternatives with a greater acuity of vision, as the cycle of innovation rolls on.