Tightly Packed Circles Hexagonal Printable A circle packing is an arrangement of circles inside a given boundary such that no two overlap and some or all of them are mutually tangent The generalization to spheres is called a sphere packing Tessellations of regular polygons correspond to particular circle packings Williams 1979 pp 35 41
This hex packing of circles is also formed when inscribing circles in the hexagons that form a regular tiling of the plane the circles are tangent at the centers of the sides of the hexagon A more complex problem is to study the packings of circles in closed surfaces when tangency patterns of the circles but not radii are prescribed Suppose I want to pack hexagons in a circle as on the drawing below red indicates packed hexagons I am wondering what is known about this problem Specifically I am interested in an approximation to how many fit given the radius of the circle and the length of the side of the hexagon and the error in such approximation
Tightly Packed Circles Hexagonal Printable
Tightly Packed Circles Hexagonal Printable
https://i.pinimg.com/originals/6d/40/a9/6d40a99e00b23d485d5385e618ecdab2.jpg
Of the circles in the packing one can equivalently search for the minimum ratio of the radius of larger circle to the radius of the circles in the packing without fixing either one The latter minimum is denoted by Dn lira One more parameter that can be optimized is the density of a packing which is the area occupied by the circles of the
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Tightly Packed Circles Hexagonal Printable
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https://en.wikipedia.org/wiki/Circle_packing
The hexagonal gaps can be filled by one circle and the dodecagonal gaps can be filled with seven circles creating 3 uniform packings The truncated trihexagonal tiling with both types of gaps can be filled as a 4 uniform packing The snub hexagonal tiling has two
https://erich-friedman.github.io/packing/cirinhex
Circles in Hexagons The following pictures show n unit circles packed inside the smallest hexagon of side length s 1 2 3 s 2 3 1 154 Trivial s 1 2 3 2 154 Trivial s 4 3 2 309 Trivial 4 5 6 s 2 3 4 7 2 666 Found by Shahriar Manzoor
https://en.wikipedia.org/wiki/Packing_problems
The hexagonal packing of circles on a 2 dimensional Euclidean plane These problems are mathematically distinct from the ideas in the circle packing theorem The related circle packing problem deals with packing circles possibly of different sizes on a surface for instance the plane or a sphere
https://nrich.maths.org/604/solution
A hexagon is 6 triangles area of 1 triangle base 2cm height 1 73cm so area of 1 triangle 1 73cm 2 and the area of the hexagon is 10 38cm 2 There are 3 whole circles in each hexagon radius of circle 1cm so area cm 2 and area of 3 circles 3 cm 2
https://plus.maths.org/content/packing-spheres
The hexagonal circle packing If the box is small then the answer depends on the shape of the box But if the box is very large the effect of the shape is negligible and the answer depends only on the volume of the box
On Circle Packing Hai Chau Chang and Lih Chung Wangyz Thue s theorem states that the regular hexagonal packing is the densest circle packing in the plane The density of this circle con guration is p 12 0 90690 In geometry circle packing refers to the study of the arrangement of unit In hexagonal packing every other row has to miss out of part of a circle from either end This creates a big waste of space When the size of the circles to be packed is large compared to the box these additional voids are significantly large enough to over rule the higher packing density
There s an intuitive way to see that d 7 is not achieved by the hexagonal packing put five disks evenly spaced rather than six around disk 0 Now add five disks into the gaps between the first five disks