doctoral candidate
Uzbekistan
A brief historical excursion into the graphics of geometry of multidimensional spaces at the paper beginning clarifies the problem – the necessary to reduce the number of geometric actions performed when depicting multidimensional objects. The problem solution is based on the properties of geometric figures called N- simplexes, whose number of vertices is equal to N + 1, where N expresses their dimensionality. The barycenter (centroid) of the N-simplex is located at the point that divides the straight-line segment connecting the centroid of the (N–1)-simplex contained in it with the opposite vertex by 1: N. This property is preserved in the parallel projection (axonometry) of the simplex on the drawing plane, that allows the solution of the problem of determining the centroid of the simplex in its axonometry to be assigned to a mechanism which is a special Assembly of pantographs (the author's invention) with similarity coefficients 1:1, 1:2, 1:3, 1:4,...1:N. Next, it is established, that the spatial location of a point in N-dimensional space coincides with the centroid of the simplex, whose vertices are located on the point’s N-fold (barycentric) coordinates. In axonometry, the ends of both first pantograph’s links and the ends of only long links of the remaining ones are inserted into points indicating the projections of its barycentric coordinates and the mechanism node, which serves as a determinator, graphically marks the axonometric location of the point defined by its coordinates along the axes х1, х2, х3 … хN.. The translational movement of the support rods independently of each other can approximate or remote the barycentric coordinates of a point relative to the origin of coordinates, thereby assigning the corresponding axonometric places to the simplex barycenter, which changes its shape in accordance with its points’ occupied places in the coordinate axes. This is an axonograph of N-dimensional space, controlled by a numerical program. The last position indicates the possibility for using the equations of multidimensional spaces’ geometric objects given in the corresponding literature for automatic drawing when compiling such programs.
simplexes, barycenter (centroid), pantograph, centroidograph of simplex, N-dimensional space, coordinate parallelepiped, baricentric coordinates, supports, axonograph of multidimensional space
1. Abdurakhmanov Sh. Pribor dlya opredeleniya tsentra tyajesti n-mernogo simpleksa. Avtorskoye svidetel’stvo SU 1031794. [Device for determining the center of gravity of an n-dimensional simplex. Author's certificate SU 1031794]. Byulleten izobreteniy SSSR [Bulletin of inventions of the USSR]. 1983, I. 28, 4 p. (in Russian)
2. Abdurakhmanov Sh. Sposoby aksonometricheskogo izobrajeniya tochek ob’yektov mnogomernykh prostranstv [Methods of axonometric image of points of objects of multidimensional spaces]. “Nauchnaya zhizn’” [Scientific life]. 2010, I. 6, pp. 50–54. (in Russian)
3. Aleksandrov P.S., Pasynkov V.A. Baritsentricheskiye koordinaty. Simpleksy [Barycentric coordinates. Simplices]. Vvedenie v teoriyu razmernosti [Introduction to dimension theory]. Moscow, Nauka Publ., 1973, pp. 196 – 199, pp. 201 – 211. (in Russian)
4. Balk M.B. Geometricheskie prilojeniya ponyatiya o tsentre tyajesti [Geometric applications of the concept of the center of gravity]. Moscow, Fizmatgiz Publ., 1959. (in Russian)
5. Balk M.B., Boltyansky V.G. Baritsentricheskie koordinaty [Barycentrische coordinates]. Geometriya mass [Geometry of the masses]. Moscow, Nauka Publ., 1987, pp. 76 – 128. (in Russian)
6. Boykov A.A. O postroyenii modeley ob’ektov prostranstva chetyryekh i bolee izmereniy v uchebnom protsesse [On constructing models of space objects of four or more of measurements in the learning process]. Geometriya i grafika [Geometry and graphics]. 2018, V. 6, I. 4, pp. 54 – 71. DOI: 10.12737/article_5c21f96dce5de8.360960 61. (in Russian)
7. Vyshnepolskiy V.I., Salkov N.A. Tseli i metody obucheniya graficheskim distsiplinam [Goals and methods of teaching graphic disciplines] Geometriya i grafika [Geometry and Graphics]. 2013, V. 1, I. 2. pp. 8 – 9. DOI: 10.12737/777. (in Russian)
8. Gordevskiy D.Z., Leybin A.S. Populyarnoe vvedenie v mnogomernuyu geometriyu [Popular introduction to multidimensional geometry]. Kharkiv, Kharkiv state University Publ., 1964, pp. 192. (in Russian)
9. Gleyzer G.I. Razvitie topologii. Obobshenie ponyatiya geometricheskogo prostranstva [Development of topology. Generalization of the concept of geometric space]. Istoriya matematiki v shkole: IX – X klassy. [History of mathematics at school: IX – X class’s]. – Moscow, Prosveshenie Publ., 1983, pp. 296 – 307. (in Russian)
10. Gumen N.S., Pavlov A.V. Zavisimost’ mezhdu elementami aksonometricheskogo proetsirovaniya v pryamougol’noy mnogomernoy aksonometrii [Dependence between elements of axonometric projection in rectangular multidimensional axonometry] Mezhvedomstvennyy respublikanskiy nauchnyy sbornik «Prikladnaya geometriya i injenernaya grafika» [Interdepartmental research collection "Applied geometry and engineering graphics"]. Kiev, Budivilnik Publ., 1967, I. 8, pp. 123 – 127. (in Russian)
11. Ivanov G.S. Predystoriya i predposylki transformatsii nachertatel’noy geometrii v inzhenernuyu geometriyu [Prehistory and prerequisites for transformation of descriptive geometry into engineering geometry]. Geometrija i grafika [Geometry and Graphics]. 2016, V. 4, I. 2, pp. 29 – 35. DOI: 10.12737/19830. (in Russian)
12. Kadykova N.S., Sal’kov N.A. Reformirovanie ocenok geometrograficheskih znaniy [The reform of the estimates of geometric and graphic knowledge]. Geometriya i grafika [Geometriy end graphiks]. 2013, V. 1, I. 2, pp. 52 – 53. – DOI: 10. 12737/475. (in Russian)
13. Kozlovskiy Yu.G. Vozniknovenie i razvitie nachertatel’noy geometrii… [Origin and development of descriptive geometry…]. Metodika kursa «Nachertatel’naya geometriya» s predisloviem prof. N.F. Chetverukhina [Technique of the course "Descriptive geometry". With a Foreword by Prof. N.F. Chetverukhin]. Minsk, Visheyshaya shkola Publ., 1971, pp. 22 – 42. (in Russian)
14. Kotov I.I. Nepolnyye chertezhi s tochki zreniya mnogomernoy geometrii [Incomplete drawings from the point of view of multidimensional geometry]. Metody nachertatel’noy geometrii i yee prilojeniya [Methods of descriptive geometry and its applications]. Moscow, State Publ. of theoretical and technical literature Publ., 1955, pp. 178–182. (in Russian)
15. Kulikov S.M. Vvedenie v nachertatel’nuyu geometriyu mnogomernykh prostranstv [Introduction to descriptive geometry of multidimensional spaces]. Moscow, Mashinostroyenie Publ., 1970, pp. 84. (in Russian)
16. Levkin Yu.S. Poluchenie chetyryokhmernykh nomogramm na baze teoremy podobiya [Getting a four dimensional nomograms on the basis of the theorems of the similarity]. Geometriya i grafika [Geometry and Graphics]. 2017, V. 5, I. 2, pp. 69–74. DOI: 10.12737/article_5953f334279642.78930109. (in Russian)
17. Levkin Yu.S. Pyatimernaya dvuxoktantovaya epyurnaya nomogramma [Five dimensional double-octante epural nomogram]. Geometriya i grafika [Geometry and Graphics]. 2017, V. 5, I. 4, pp. 44–51. DOI: 10.12737/article_5a17fecf 2feac9.1812 3975. (in Russian)
18. Levkin Yu.S. Pyatimernaya dvuxoktantovaya epyurnaya nomogramma [Six dimensional epural nomogram in four-octante measurement]. Geometriya i grafika [Geometry and Graphics]. 2018, V. 6, I. 1, pp. 39 – 47. DOI: 10.12737/article_5ad098b0 5f l559.36303938. (in Russian)
19. Lyashkov A.A., Panchuk K.L., Varelo L.G. Osobennost’ otobrazheniya giperpoverkhnosti chetyrexmernogo prostranstva [Feature of displaying the Hyper –surface of a four-dimensional space]. Geometriya i grafika [Geometry and Graphics]. 2013, V. 5, I. 3, pp. 3 – 10. DOI: 10.12737/article_59bfa3078af4c1.45321238. (in Russian)
20. Manevich V.A., Kotov I.I., Zengin A.R. Tochechnye massy i vektory. Baritsentricheskie koordinaty [Point masses and vectors. Barycentric coordinates]. Analiticheskaya geometriya s teoriyey izobrajeniy [Analytical geometry with image theory]. Moscow, Vishaya shkola Publ., 1969, pp. 7–20. (in Russian)
21. Myakishev A.G. Baritsentricheskie koordinaty [Barycentric coordinates]. Elementy geometrii treugol’nika [Elements of geometrics of a triangle]. Moscow, Center for continuous mathematical education Publ., 2002, pp. 17–28. (in Russian)
22. Narzullaev S.A. Raschet n-mernogo prostranstva [Calculation of n-dimensional space]. Tashkent, "Fan" Publ., 1991. p. 224. (in Russian)
23. Naumovich N.V. Primeneniye mnogomernoy nachertatel’noy geometrii k dokazatel’stvu nekotoryx teorem planimetrii i stereometrii [Application of multidimensional descriptive geometry to the proof of some theorems of planimetry and stereometry]. Metody nachertatel’noy geometrii i yee prilojeniya [Methods of descriptive geometry and its applications]. Moscow, State Publ. of theoretical and technical literature, 1955, pp. 183–193. (in Russian)
24. Pervikova V.N. Obobshenie osnovnoy teoremy tsentral’noy aksonometrii na prostranstvo n izmereniy [Generalization of the main theorem of Central axonometry on the space of n dimensions]. Metody nachertatel’noy geometrii i yee prilojeniya [Methods of descriptive geometry and its applications]. Moscow, State Publ. of theoretical and technical literature, 1955, pp. 141–155. (in Russian)
25. Pryanishnikova Z.I. Obobshenie proektsiy Ye. S. Fedorova [Generalization of projections by E. S. Fedorov] Metody nachertatel’noy geometrii i yee prilojeniy [Methods of descriptive geometry and its applications]. Moscow, State Publ. of theoretical and technical literature, 1955, pp.156 – 174.
26. Pukhnachev Yu.V., Popov Yu.P. Puteshestvie po vsem prostranstvam [Journey through all the spaces]. Yunyy texnik [Young technician]. 1967, I. 12, pp. 42 – 45. (in Russian)
27. Rosenfeld B.A. Mnogomernye prostranstva [Multidimensional spaces]. Moscow, Nauka Publ., 1966. p. 648. (in Russian)
28. Salkov N.A. Otobrajenie problem geometricheskogo obrazovaniya v jurnale «Geometriya i grafika» [The image of the problem of geometric education in the journal “Geometry and graphics”]. Geometriya i grafica [Geometry and Graphics]. 2020, V. 8, I. 3, pp. 87 – 119. DOI: 10.12737/2308-4898-2020-87-119. (in Russian)
29. Seliverstov A.V. Prikladnaya mnogomernaya nachertatel’naya geometriya segodnya i vchera [Applied multidimensional descriptive geometry today and yesterday]. Materialy VIII mejdunarodnoy internet-konferentsii “Kachestvo graficheskoy podgotovki” [Materials of the VIII international Internet conference “quality of graphic training”]. Moscow, 2019. (in Russian)
30. Sokolova L.S. Abstraktnye mnogomernye prostranstva v injenernom obrazovanii [Abstract multidimensional spaces in engineering education]. Materialy VIII mezhdunarodnoy internet-konferentsii “Kachestvo graficheskoy podgotovki” [Materials of the VIII international Internet conference “Quality of graphic training”]. Moscow, 2019. (in Russian)
31. Sokolova L.S. Mnogomernoe prostranstvo i naglyadnaya geometriya v uchebnoy programme po geometricheskoy podgotovke dlya bakalavriata [Multidimensional space and visual geometry in the curriculum for geometric training for undergraduate students] Geometriya i grafika [Geometry and Graphics]. 2015, V. 3, I. 1, pp. 40–46. DOI: 10.12737/10457. (in Russian)
32. Filippov P.V. Nachertatel’naya geometriya mnogomernogo prostranstva i yee prilojeniya [Descriptive geometry of multidimensional space and its applications]. Leningrad, LGU Publ., 1979, pp. 280. (in Russian)
33. Chetverukhin N.F Glazunov E.A. Mnogomernaya aksonometriya [Multidimensional axonometry]. Aksonometriya [Axonometry]. Moscow, State Publ. of theoretical and technical literature, 1953, pp. 72 – 87. (in Russian)
34. Chetverukhin N.F. O nekotorykh voprosakh mnogomernoy geometrii [On some questions of multidimensional geometry]. Mejvedomstvennyy respublikanskiy nauchnyy sbornik “Prikladnaya geometriya i injenernaya grafika” [Interdepartmental Republican scientific collection "Applied geometry and engineering graphics"]. Kiev, Budivilnik Publ., 1970, I. 10, pp.10 – 12. (in Russian)
35. Chetverukhin N.F. Formy vysshix stupeney v mnogomernom rasshirennom yevklidovom prostranstve [Forms of higher stages in a multidimensional extended Euclidean space]. Mejvedomstvennyy respublikanskiy nauchnyy sbornik “Prikladnaya geometriya i injenernaya grafika” [Interdepartmental Republican scientific collection "Applied geometry and engineering graphics"]. Kiev, Budivilnik Publ., 1971, I. 12, pp. 3 – 4. (in Russian)
36. Shitov V.M. Aksonografy. Avtomaticheskiye chertezhnye mashiny [Axonographs. Automatic drawing machines]. Mexanizatsiya chertezhno-konstruktorskikh rabot [Mechanization of drawing and design works]. Moscow, Mashinostroyenie Publ., 1966, pp. 31–33, pp. 47–52. (in Russian)
37. Yurkov V.Yu. Formal’noe predstavlenie usloviy intsidentnosti v mnogomernykh proektivnykh prostranstvakh [Formal representation of incident conditions in multidimensional projective spaces]. Geometriya i grafika [Geometry and Graphics]. 2016, V. 4, I. 4, pp. 3 – 13. DOI: 10.12737/22838. (in Russian)
38. Möbius, August Ferdinand. Der baricentrishe Calcul. Leipzig, 1827, pp. 460.
