Algorithm Optimisation Championship. - page 19
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You should at least read some books . At least Penrose, The New King's Mind, for the sake of an outlook, read one book...
Maybe you should start with a basic course in geometry. What is a point and how many dimensions it takes. What is a segment, a line, how many dimensions they take up. Move on to volumetric shapes. From simple to complex, step by step.
Understand that we should not limit ourselves to what our senses can sense and measure, the world is much more vast and immense to be measured in three dimensions.
Andrew, with all due respect, I will not have time to read Penrose before the championship.
But my question is: why is there no clarity of the problem?
You talk about multidimensionality of space, but you yourself say that you can't represent a surface in it (see the quote above).
I KNOW FROM MY HIGH SCHOOL GEOMETRY CURRICULUM THAT ANY POINT IN SPACE IS IN THREE DIMENSIONS.
A point is positioned in space using X,Y and Z coordinates, where each axis represents one dimension of the three dimensional space.
A plane is a space of two coordinates, X and Y. Where X is the horizontal axis and Y is the vertical axis.
No physical body (point) can go beyond the X,Y,Z coordinate axes.
Mathematically, - a point can exist in two-dimensional space, - in the plane of a drawn graph.
Physically, - a point can exist in at least three dimensions and not less.
Our FF function is mathematical. SO IT DOES NOT REQUIRE MORE THAN THREE DIMENSIONS FOR ITS CURVE. You said it yourself - FF is an analytic function.
The school curriculum, in analytic geometry, tells without unnecessary complications how curves are constructed in a graph by means of points whose coordinates are calculated in the equation of the function.
If our FF is an analytic function - then it also returns coordinates of points on a graph. If we connect these points with a line, we get a curve. This curve has its low and high points.
I understood the problem this way: we need to optimize the search for the upper points (maximums) of the unknown analytic function. (which on the graph will just look like a curved line).
Simplified, I understood search optimization as the development of an algorithm that allows to get rid of the need to streamline the curve to find vertices in the graph (which means a complete enumeration of all values passed in the equation of the analytic function), and relying on the logic of the minimum number of available coordinates, find peaks of this curve in the graph.
See where I got the curved line and surface analogy from. https://www.mql5.com/ru/forum/84457/page3
Here I think I'm back... :)
Andrei, with all due respect, I won't have time to read Penrose before the championship starts.
But my question is: why there is no clarity of the problem?
You talk about multidimensionality of space, but you yourself say that you cannot represent a surface in it (see the quote above).
I KNOW FROM SCHOOL GEOMETRY CURRICULUM THAT ANY POINT IN SPACE IS IN THREE DIMENSIONS.
A point is positioned in space using the coordinates of the X,Y,Z axes, where each axis represents one dimension of three-dimensional space.
A plane represents a space of two coordinates, X and Y. Where X is the horizontal axis and Y is the vertical axis.
No physical body (point) can go beyond the X,Y,Z coordinate axes.
Mathematically, - a point can exist in two-dimensional space, - in the plane of a drawn graph.
Physically, - a point can exist in at least three dimensions and not less.
Our FF function is mathematical. SO IT DOES NOT REQUIRE MORE THAN THREE DIMENSIONS FOR ITS CURVE. You said it yourself - FF is an analytic function.
The school curriculum, in analytic geometry, tells you without too much complication how curves are constructed on a graph using points whose coordinates are calculated in the function equation.
If our FF is an analytic function - then it also returns coordinates of points on a graph. If we connect these points with a line, we get a curve. This curve has its low and high points.
I understood the problem this way: we need to optimize the search for the upper points (maximums) of the unknown analytic function. (which on a graph will just look like a curve line).
To simplify, I understood search optimization as development of algorithm which allows to get rid of necessity of pointwise reproduction of curve to find tops on graph (which means complete search of all values of analytic function in equation), and relying on logic of minimum quantity of available coordinates to find peaks of this curve on the graph.
I don't know why you don't have clarity of the problem. But I can make a guess - because you have several errors in your reasoning. For example, you confuse "the required number of measurements to construct an object" and "the number of measurements in which the object is located".
I don't know why you don't have clarity of purpose. But I can make a guess - because you have some errors in your reasoning. For example, you confuse "the required number of measurements to construct an object" and "the number of measurements in which the object is located".
Well, why do I confuse it...
Look here:
An object is a curved line drawn on a graph by drawing one line through n points whose coordinates are obtained by solving levels of some analytic function.
Necessary number of measurements to construct an object: - Determined by calculating the coordinates of the minimum number of points on the plane (or in space) of a graph, for the subsequent drawing of a line through them. The coordinate calculations need exactly as many measurements as the curved line we need.
It dependson whether the curve line is drawn in plane or space. If on a plane, the object curved line, will be in two dimensions - Height and Length, represented by the X and Y coordinate axes. If we draw a curved line that goes through space (such as inside a cube), the number of measurements of the object will increase, so as to have to calculate the coordinates of the object in one more dimension - Width, represented by the Z-axis. In total, there will be three dimensions X,Y,Z . (Of course the analytic function itself has to return the Z-axis coordinates).
The analytical function, is simply a mathematical equation that represents the spatial phenomenon of the surface of various geometric objects. It provides the full range of coordinates needed to construct various curved lines. However, the more complex the line, the more complex the equation that returns its coordinates on the graph.
Any geometric body can be any number of dimensions. In one-dimensional space a segment, in two-dimensional the same object is a rectangle, in three-dimensional a cube, in four-dimensional a hypercube, etc. there is no limit.
Any geometric body can be at least as peaceful. In one-dimensional space a segment, in two-dimensional the same object is a rectangle, in three-dimensional a cube, in four-dimensional a hypercube, etc. there is no limit.
Any geometric body can be any number of dimensions. In one-dimensional space a segment, in two-dimensional the same object is a rectangle, in three-dimensional a cube, in four-dimensional a hypercube, etc. there is no limit.
Well, if we are going to base championship rules on such theories, then academics may join our competition and you and I will risk "sitting in a puddle" :)
I've already written that there's no need to get hung up on the representation of multidimensional spaces. A function can have any number of parameters - obviously, plain and simple. And to represent exactly two-dimensional graph and three-dimensional graph, look for maximum or minimum on them. All the rest must be done by the correct approach in programming: a parameter defining the number of parameters, dynamic arrays in accordance with this number, loops repeated in accordance with this parameter.
Limit yourself to one or two optimizable parameter, but make it work automatically, only by setting property, defining number of parameters. And from there, you can slip any number of parameters in.
You began to enumerate "dimensions" of geometrical bodies so confidently, that I already thought, you will continue and begin to enumerate other, unknown to me dimensions, but you stopped at the fourth known dimension. Time. Please continue your list of dimensions. :)
...5-dimensional, 6-dimensional, 7-dimensional, 8-dimensional, 9-dimensional, 10-dimensional, 11-dimensional, 12-dimensional...
More?