GB2593463A - A method for the construction of densest helical structures of equal-sized spheres in cylindrical confinement - Google Patents
A method for the construction of densest helical structures of equal-sized spheres in cylindrical confinement Download PDFInfo
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- GB2593463A GB2593463A GB2004138.0A GB202004138A GB2593463A GB 2593463 A GB2593463 A GB 2593463A GB 202004138 A GB202004138 A GB 202004138A GB 2593463 A GB2593463 A GB 2593463A
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- 238000000034 method Methods 0.000 title claims abstract description 14
- 238000010276 construction Methods 0.000 title claims abstract description 9
- 238000013461 design Methods 0.000 claims abstract description 7
- 238000011160 research Methods 0.000 claims abstract description 4
- 238000000926 separation method Methods 0.000 claims description 12
- 230000001419 dependent effect Effects 0.000 abstract 1
- 238000012856 packing Methods 0.000 description 2
- 239000013078 crystal Substances 0.000 description 1
- 238000012986 modification Methods 0.000 description 1
- 230000004048 modification Effects 0.000 description 1
- 230000000737 periodic effect Effects 0.000 description 1
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
- G06F17/10—Complex mathematical operations
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
- G06F17/10—Complex mathematical operations
- G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
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Abstract
A method for constructing densest helical structures of equal-sized spheres in cylindrical confinement, comprising single-helix structures and double-helix structures. The method comprises mathematical formulae for the construction of densest single-helix structures for values of D in the range (1+√3/2, 1+4√3/7) and densest double-helix structures for values of D in the range (1+4√3/7, 2), where D is diameter of the cylinder divided by the diameter of the spheres. The formulae may be used to determine the exact positions of the spheres in order to construct the single and double-helix structures in professions such as research, product design, art and architecture. The formulae give the positional coordinates of the nth sphere in the densest single-helix or double-helix structure using cylindrical coordinates, wherein each coordinate is dependent only on n and D.
Description
DESCRIPTION
A Method for the Construction of Densest Helical Structures of Equal-sized Spheres in Cylindrical Confinement
Technical Field
[0001] The invention is related to the technical field of sphere packing in cylindrical confinement. In particular, it refers to a method for the construction of densest helical structures of equal-sized spheres in cylindrical confinement.
Background Art
[0002] The periodic structures of densest packings of equal-sized hard spheres in cylindrical confinement are referred to as columnar crystals, where such structures depend only on the ratio D = cylinder diameter / sphere diameter. In the past two decades, a variety of structures have been discovered computationally for different regimes of D. Apart from the zigzag structures at D <1+ VI, all other densest structures were only discovered numerically without any mathematical formula for the exact positions of spheres.
Summary of the Invention
[0003] The invention provides a method for the construction of densest helical structures of equal-sized spheres in cylindrical confinement, in particular for the construction of densest single-helix structures at 1) E (1+ V172,1+ 411/7) and densest double-helix structures at 1) E (1+ 4-11/7, 2) , wherein = cylinder diameter / sphere diameter.
DESCRIPTION
[0004] As a further description of the invention, for any densest single-helix structure at E (1+ -1.V2,1+ 4^13/7) , wherein D = cylinder diameter / sphere diameter, the angular separation AO and vertical separation Az of any consecutive pair of spheres are given by AO = arccos[l --r3/(/) -1)1 Eq. (1) and Az =11-(D -1)/2 Eq. (2) respectively, such that, for integers n = 1, 2, 3, 4, the positional coordinates of the nth sphere in the densest single-helix structure are given by r= (D -1)/2 Eq. (3) z = (n -1)Az [0005] As a further description of the invention, for any densest double-helix structure at D E (1+477,2) , wherein D = cylinder diameter / sphere diameter, the angular separation AO and vertical separation Az of any consecutive pair of spheres are given by Eq. (4) 3(D -1)±18-811-(1) -02 _ 7(D -02 Aر = arccos 2(D -1)( 1 -1 1 --1)2) and 3(D-1)2 ± (/) -1)18 -811-(I) --7(D-lf 4(1-11-(1)-1)2) Eq. (5)
DESCRIPTION
respectively, such that, for integers ii - 3 densest double-helix structure are given by.., the positional coordinates of the nth sphere in the r = -1)12 Eq. (6), 0 =[nI2]\0-+ [01 -1)/21A 0+ = [n12]Az +[(n -1)12]Az' wherein the square bracket represents a rounding down to the closest integer as illustrated by the following table: a [7/12] [(n -1)/2] 1 0 0 2 1 0 3 1 1 4 2 1 5.2, 2 6 3 2 7 3 3 8 4 3 9 4 4 5 4 [0006] As a further description of the invention, Eqs. (1), (2) and (3), or their equivalents, can be used to construct any densest single helix structure at D e (1+ 12I + 4%h/7).
[0007] As a further description of the invention, Eqs. (4), (5) and (6), or their equivalents, can be used to construct any densest double helix structure at D E (1 + 4-NR/7,2).
DESCRIPTION
[0008] Mathematical formulae for the exact positions of spheres have been obtained for the densest single-helix structures at 1) E ± , 1 ± 4177) and the densest double-helix structures at D E ± 473/7, 2) so that anyone who needs to construct such structures in his/her own profession (e.g. research, product design, art, and architecture) can use our mathematical formulae to determine the exact positions of spheres.
Brief Description of Accompanying Drawings
[0009] FIG. 1 illustrates a densest single-helix structure at D E (1+-Z/2,1+ z1-15/7) for the invention; [0010] FIG. 2 illustrates a densest double-helix structure at D E (1 ± 4.11/7,2) 2) for the invention, [0011] FIG 3 is a table that illustrates how parameters inside the square brackets of Eq. (6) are rounded down to their closest integers.
Detailed Description of the Preferred Embodiments
[0012] The invention will be further described below in details with reference to the accompanying drawings.
[0013] Embodiment 1: As illustrated by FIGs. 1, 2 and 3, the invention provides a method for the construction of densest helical structures of equal-sized spheres in cylindrical confinement, in particular for the construction of densest single-helix structures at D E (1+ VV2,1+ zkh/7) and densest double-helix structures at D E + 4f3/7, 2) where D = cylinder diameter / sphere diameter.
DESCRIPTION
[0014] As illustrated by FIG. 1, for any densest single-helix structure at D E (1+172,1+ 411/7), where D = cylinder diameter / sphere diameter, the angular separation AO and vertical separation Az of any consecutive pair of spheres are given by AO = arccosP -,1731(D -1)] Eq. (1) and Az = ,h(D -1)/2 Eq. (2) respectively, such that, for integers n = 1, 2, 3, 4, the positional coordinates of the nth sphere in the densest single-helix structure are given by r= (D -1)/2 z = (n -1) Az Eq. (3) [0015] As illustrated by FIG. 2, for any densest double-helix structure at D E (1+ 4N/S77, 2) , where = cylinder diameter / sphere diameter, the angular separation AO and vertical separation A7 of any consecutive pair of spheres are given by 3(D -0-48 - 02 _ 7(D-02 2(D -1)(1--11-(D-1)2) 3(D -1)' ± -1 8 - -_ 7(D-02 4(1--(D-1)2) AØ ± = arccos and Azi = respectively, such that, for integers n = 1, 2, 3, 4...,the positional coordinates of the nth sphere in the densest double-helix structure are given by Eq. (4.) Eq. (5)
DESCRIPTION
= (D -1)/2 0 = [n/2]A0 + [(ri -1)12]A0' Eq. (6), z = [n12]Az + kn -1)/ 2117 wherein the square bracket represents a rounding down to the closest integer as illustrated by the table in FIG. 3.
[0016] As a further description of the invention, Eqs. (1), (2) and (3), or their equivalents, can be used to construct any densest single helix structure at 1) E (1±-j3/2,1+44177).
[0017] As a further description of the invention, Eqs (4), (5) and (6), or their equivalents, can be used to construct any densest double helix structure at D e (I + 4-13/7, 2) [0018] Mathematical formulae for the exact positions of spheres have been obtained for the densest single-helix structures at D E ±VY2,1+41V7) and the densest double-helix structures at D e (1+ 413/7, 2) , so that anyone who needs to construct such structures in his/her own profession (e.g. research, product design, art, and architecture) can use our mathematical formulae to determine the exact positions of spheres.
[0019] For anyone regardless of his/her own profession (e.gresearch, product design, art, and architecture), the following action must require permission from the patent owners: For Eqs. (1), (2) and (3), use one or more of these equations (or their equivalents) to construct any densest single-helix structure at D E (1+13/2,1 + 415/7).
DESCRIPTION
[0020] For anyone regardless of his/her own profession (e.gresearch, product design, art, and architecture), the following action must require permission from the patent owners: For Eqs. (4), (5) and (6) use one or more of these equations (or their equivalents) to construct any densest double-helix structure at D e (1+441/7,2) . [0021] For anyone regardless of his/her own profession (e.gresearch, product design, art, and architecture), the following action must require permission from the patent owners: Use any equation or related contents in the publication [AlP Advances 9, 125118 (2019), hap s://c161 or,g/10.1 003/1.5131318] to construct any densest single-helix structure at D E ± V3/2, 1 413/7) or any densest double-helix structure at D E ± 4-shh, 2).
[0022] The basic principles and main features of the invention are described above, and it should be understood by those skilled in the art that the invention is not limited by the foregoing embodiments. While the above embodiments and specifications describe only the principles of the invention, various modifications and improvements of the invention can be made without departing from the scope of the invention, which are within the scope of the invention as claimed. The scope of the invention is defined by the appended claims and their equivalents.
Claims (7)
- CLAIMS1. A method for constructing densest helical structures of equal-sized spheres in cylindrical confinement, comprising single-helix structures and double-helix structures.
- 2. The method according to claim 1, further comprising mathematical formulae for the construction of densest single-helix structures at D E (1+ V5/2,1+ 45/7) and densest double-helix structures at D e (1+ 4f3/7, 2) , wherein D = cylinder diameter / sphere diameter.
- 3. The method according to claim 2, anyone who needs to construct the single-helix and double-helix structures in his/her own profession (e.g. research, product design, art, and architecture) can use the mathematical formulae elicited in the invention to determine the exact positions of spheres.
- 4. The method according to claim 2, for any densest single-helix structure at D E (1+ ^172,1+ 4ji/7) , wherein D = cylinder diameter / sphere diameter, the angular separation AO and vertical separation A7 of any consecutive pair of spheres are given by AO = arccos [1 -VIAL) -1)] Eq. (I) and A7 = V1 -,h(D -1)/2 Eq. (2) respectively, such that, for integers ri = 1,2, 3,4 the positional coordinates of the nth sphere in the densest single-helix structure are given by r = (D -1)/2 =(n-1)A0 z = (n -1)A7 Eq. (3) The method according to claim 2, for any densest double-helix structure at D E (1+ 4.11/7,2) 2) , wherein D = cylinder diameter / sphere diameter, the angular separation AO and vertical separation Az of any consecutive pair of spheres are given byCLAIMS
- -1)± -8V1-0 -02 - -1)2 2W-1) 1-111-(D-iy A01-= arecos Eq. (4) and 3(D-1)2 ±(D 48 -8 11-(D --7(D -1 4(1--(D -) Eq. (5,) respectively, such that, for integers n - .., the positional coordinates of the nth sphere in the densest double-helix structure are given by r = (D -1)/2 0 = [71/2]z\0-kn -1)121\0+ z =[n 12]Az-+[(n -1)12]Az+ Eq. (6), wherein the square bracket represents a rounding down to the closest integer.
- 6 The method according to claim 4, Eqs (1), (2) and (3), or their equivalents, can be used to construct any densest single-helix structure at D E (1+ V72,1+ 4N5/7) .
- 7. The method according to claim 5 Eqs. (4), (5) and (6), or their equivalents can be used to construct any densest double-helix structure at D E (1+ 4,11/7,2)
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201910220723.5A CN109948261B (en) | 2019-03-22 | 2019-03-22 | Method for constructing spiral closest packing structure of equal-volume sphere in circular tube |
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| Publication Number | Publication Date |
|---|---|
| GB202004138D0 GB202004138D0 (en) | 2020-05-06 |
| GB2593463A true GB2593463A (en) | 2021-09-29 |
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| Application Number | Title | Priority Date | Filing Date |
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| GB2004138.0A Withdrawn GB2593463A (en) | 2019-03-22 | 2020-03-23 | A method for the construction of densest helical structures of equal-sized spheres in cylindrical confinement |
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| GB (1) | GB2593463A (en) |
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| JP2514304B2 (en) * | 1993-08-04 | 1996-07-10 | 出光石油化学株式会社 | Polymer-granular body and method for producing the same |
| CN100457609C (en) * | 2000-11-13 | 2009-02-04 | 国际商业机器公司 | Manufacturing method and application of single wall carbon nano tube |
| KR101585286B1 (en) * | 2009-03-13 | 2016-01-13 | 하마마츠 포토닉스 가부시키가이샤 | Radiation image conversion panel and method for producing same |
| CN109202270B (en) * | 2017-11-24 | 2021-06-08 | 中国航空制造技术研究院 | Double-helix stirring method and stirring device in additive manufacturing |
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- 2019-03-22 CN CN201910220723.5A patent/CN109948261B/en active Active
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- 2020-03-23 GB GB2004138.0A patent/GB2593463A/en not_active Withdrawn
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| Publication number | Publication date |
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| CN109948261A (en) | 2019-06-28 |
| GB202004138D0 (en) | 2020-05-06 |
| CN109948261B (en) | 2023-04-07 |
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