Abstract
In this paper, we present our research on building computing machines consciousness about intuitive geometry based on mathematics experiments and statistical inference. The investigation consists of the following five steps. At first, we select a set of geometric configurations and for each configuration we construct a large amount of geometric data as observation data using dynamic geometry programs together with the pseudo-random number generator. Secondly, we refer to the geometric predicates in the algebraic method of machine proof of geometric theorems to construct statistics suitable for measuring the approximate geometric relationships in the observation data. In the third step, we propose a geometric relationship detection method based on the similarity of data distribution, where the search space has been reduced into small batches of data by pre-searching for efficiency, and the hypothetical test of the possible geometric relationships in the search results has be performed. In the fourth step, we explore the integer relation of the line segment lengths in the geometric configuration in addition. At the final step, we do numerical experiments for the pre-selected geometric configurations to verify the effectiveness of our method. The results show that computer equipped with the above procedures can find out the hidden geometric relations from the randomly generated data of related geometric configurations, and in this sense, computing machines can actually attain certain consciousness of intuitive geometry as early civilized humans in ancient Mesopotamia.
Highlights
Intuitive geometric knowledge is an origin of human civilization, just as shown by the Plimpton 322 tablet that people in the Old Babylonian period already knew the rule of the right triangle i.e., the Pythagorean theorem, through various instances of right triangles, almost one thousand years before proof was given in Greek time
We present our research on building computing machines consciousness about intuitive geometry based on mathematics experiments and statistical inference
The results show that computer equipped with the above procedures can find out the hidden geometric relations from the randomly generated data of related geometric configurations, and in this sense, computing machines can attain certain consciousness of intuitive geometry as early civilized humans in ancient Mesopotamia
Summary
Intuitive geometric knowledge is an origin of human civilization, just as shown by the Plimpton 322 tablet that people in the Old Babylonian period (between −1900 and −1600) already knew the rule of the right triangle i.e., the Pythagorean theorem, through various instances of right triangles, almost one thousand years before proof was given in Greek time. The research work of computer proof of geometric theorems is mainly developed from the following three directions: 1) Algebraic calculation method based on coordinates; 2) Point elimination method based on geometric invariants; 3) Proving theorems by simulating human thinking the reasoning database search method. When the above-mentioned algebraic methods are used to prove geometric theorems, they usually include large-scale complicated calculations involving polynomials, which geometric meaning generally can’t be understood by human, and for human it is too difficult to check the correctness of the machine computation by manual method.
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