Novel Recurrence Relations for Volumes and Surfaces of <i>n</i>-Balls, Regular <i>n</i>-Simplices, and <i>n</i>-Orthoplices in Real Dimensions
This study examines <i>n</i>-balls, <i>n</i>-simplices, and <i>n</i>-orthoplices in real dimensions using novel recurrence relations that remove the indefiniteness present in known formulas. They show that in the negative, integer dimensions, the volumes of <i&...
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| Format: | Article |
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MDPI AG
2022-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/13/2212 |
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| author | Szymon Łukaszyk |
| author_facet | Szymon Łukaszyk |
| author_sort | Szymon Łukaszyk |
| collection | DOAJ |
| description | This study examines <i>n</i>-balls, <i>n</i>-simplices, and <i>n</i>-orthoplices in real dimensions using novel recurrence relations that remove the indefiniteness present in known formulas. They show that in the negative, integer dimensions, the volumes of <i>n</i>-balls are zero if <i>n</i> is even, positive if <i>n</i> = −4<i>k</i> − 1, and negative if <i>n</i> = −4<i>k</i> − 3, for natural <i>k</i>. The volumes and surfaces of <i>n</i>-cubes inscribed in <i>n</i>-balls in negative dimensions are complex, wherein for negative, integer dimensions they are associated with integral powers of the imaginary unit. The relations are continuous for <i>n</i> ∈ ℝ and show that the constant of <i>π</i> is absent for 0 ≤ <i>n</i> < 2. For <i>n</i> < −1, self-dual <i>n</i>-simplices are undefined in the negative, integer dimensions, and their volumes and surfaces are imaginary in the negative, fractional ones and divergent with decreasing <i>n</i>. In the negative, integer dimensions, <i>n</i>-orthoplices reduce to the empty set, and their real volumes and imaginary surfaces are divergent in negative, fractional ones with decreasing <i>n</i>. Out of three regular, convex polytopes present in all natural dimensions, only <i>n</i>-orthoplices and <i>n</i>-cubes (and <i>n</i>-balls) are defined in the negative, integer dimensions. |
| first_indexed | 2024-03-09T04:02:10Z |
| format | Article |
| id | doaj.art-16df770d99fd4bc8a697ba94f8dad5d3 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T04:02:10Z |
| publishDate | 2022-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-16df770d99fd4bc8a697ba94f8dad5d32023-12-03T14:11:47ZengMDPI AGMathematics2227-73902022-06-011013221210.3390/math10132212Novel Recurrence Relations for Volumes and Surfaces of <i>n</i>-Balls, Regular <i>n</i>-Simplices, and <i>n</i>-Orthoplices in Real DimensionsSzymon Łukaszyk0Łukaszyk Patent Attorneys, ul. Głowackiego 8, 40-052 Katowice, PolandThis study examines <i>n</i>-balls, <i>n</i>-simplices, and <i>n</i>-orthoplices in real dimensions using novel recurrence relations that remove the indefiniteness present in known formulas. They show that in the negative, integer dimensions, the volumes of <i>n</i>-balls are zero if <i>n</i> is even, positive if <i>n</i> = −4<i>k</i> − 1, and negative if <i>n</i> = −4<i>k</i> − 3, for natural <i>k</i>. The volumes and surfaces of <i>n</i>-cubes inscribed in <i>n</i>-balls in negative dimensions are complex, wherein for negative, integer dimensions they are associated with integral powers of the imaginary unit. The relations are continuous for <i>n</i> ∈ ℝ and show that the constant of <i>π</i> is absent for 0 ≤ <i>n</i> < 2. For <i>n</i> < −1, self-dual <i>n</i>-simplices are undefined in the negative, integer dimensions, and their volumes and surfaces are imaginary in the negative, fractional ones and divergent with decreasing <i>n</i>. In the negative, integer dimensions, <i>n</i>-orthoplices reduce to the empty set, and their real volumes and imaginary surfaces are divergent in negative, fractional ones with decreasing <i>n</i>. Out of three regular, convex polytopes present in all natural dimensions, only <i>n</i>-orthoplices and <i>n</i>-cubes (and <i>n</i>-balls) are defined in the negative, integer dimensions.https://www.mdpi.com/2227-7390/10/13/2212regular convex polytopesnegative dimensionsfractal dimensionscomplex dimensions |
| spellingShingle | Szymon Łukaszyk Novel Recurrence Relations for Volumes and Surfaces of <i>n</i>-Balls, Regular <i>n</i>-Simplices, and <i>n</i>-Orthoplices in Real Dimensions Mathematics regular convex polytopes negative dimensions fractal dimensions complex dimensions |
| title | Novel Recurrence Relations for Volumes and Surfaces of <i>n</i>-Balls, Regular <i>n</i>-Simplices, and <i>n</i>-Orthoplices in Real Dimensions |
| title_full | Novel Recurrence Relations for Volumes and Surfaces of <i>n</i>-Balls, Regular <i>n</i>-Simplices, and <i>n</i>-Orthoplices in Real Dimensions |
| title_fullStr | Novel Recurrence Relations for Volumes and Surfaces of <i>n</i>-Balls, Regular <i>n</i>-Simplices, and <i>n</i>-Orthoplices in Real Dimensions |
| title_full_unstemmed | Novel Recurrence Relations for Volumes and Surfaces of <i>n</i>-Balls, Regular <i>n</i>-Simplices, and <i>n</i>-Orthoplices in Real Dimensions |
| title_short | Novel Recurrence Relations for Volumes and Surfaces of <i>n</i>-Balls, Regular <i>n</i>-Simplices, and <i>n</i>-Orthoplices in Real Dimensions |
| title_sort | novel recurrence relations for volumes and surfaces of i n i balls regular i n i simplices and i n i orthoplices in real dimensions |
| topic | regular convex polytopes negative dimensions fractal dimensions complex dimensions |
| url | https://www.mdpi.com/2227-7390/10/13/2212 |
| work_keys_str_mv | AT szymonłukaszyk novelrecurrencerelationsforvolumesandsurfacesofiniballsregularinisimplicesandiniorthoplicesinrealdimensions |