I am trying to understand the differences between the Honeybee Conjecture and the Sphere Packing Conjecture (also called the Kepler Conjecture).
As a quick overview:
"The honeycomb conjecture states that a regular hexagonal grid or honeycomb is the best way to divide a surface into regions of equal area with the least total perimeter." (https://en.wikipedia.org/wiki/Honeycomb_conjecture)
"The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. The density of these arrangements is around 74.05%." (https://en.wikipedia.org/wiki/Kepler_conjecture)
Reading this, I had the following questions:
1) The Honeycomb Conjecture says that a hexagonal grid is the best way to divide regions into equal area and the least perimeter, whereas the Sphere Packing Conjecture says that closest packing arrangements can be achieved EITHER using Cubic Arrangements OR Hexagonal Arrangements. Do the results from both these conjectures contradict each other - or are both these conjectures addressing fundamentally different problems?
2) From the Wikipedia Page on the Honeycomb Conjecture, (apparently) the official mathematical formulation of the Honeycomb Conjecture can be written as follows:
Why do we want the supremum of the limit of "r" to approach infinity? Why are we interested in the perimeter and area of the "bounded components and the disk of radius r" - does this represent a choice of tiling arrangement? Would the disk B(0,r) represent a choice of tiling like a hexagon? The fourth root of 12 is 1.86 : why is 1.86 relevant here?
3) Finally, for both the Honeybee Conjecture and the Sphere Packing Conjecture - are we able to "bound" the sample space of the solution (i.e. how many possible arrangements can exist)?
For instance, suppose we want to predict the the possible outcomes that can arise from flipping a two-sided coin twice - the sample space corresponding to this problem would include 4 outcomes : (Head, Head), (Head, Tail), (Tail, Head), (Tail, Tail)
If we were to consider the Sphere Packing Conjecture, we know that the sample space to this problem contains at least 3 possible arrangements: Cubic, Hexagonal and Random. But do there exist more possible arrangements? Are the number of total possible arrangements finite?
If we were to consider the Honeybee Conjecture, the following video (https://www.youtube.com/watch?v=7edkFs8Vu1E @ 4:03 ) shows us that there the sample space of this problem contains at least 5 possible arrangements:
But do there exist more possible arrangements? Are the number of total possible arrangements finite?
I found this Wikipedia Page (https://en.wikipedia.org/wiki/Uniform_tiling) that contains (beautiful) pictures about different ways to "partition and pack" space:
Reading the Wikipedia page, the Euclidean Plane seems to have 3 types of uniform tiling arrangements : Square, Hexagonal and Triangular. But reading the "Uniform Tilings Using Star Polygons" section - it seems like you can have plenty of other types of uniform tiling arrangements (e.g. Topological 3.12.12 , Topological 184.108.40.206, , Topological 6.6.6, etc.).
"The complete lists of k-uniform tilings have been enumerated up to k=6. There are 20 2-uniform tilings, 61 3-uniform tilings, 151 4-uniform tilings, 332 5-uniform tilings, and 673 6-uniform tilings." (https://en.wikipedia.org/wiki/List_of_k-uniform_tilings) - Is it possible there could existing more tiling arrangements for k>6?
Do there exist finite or infinite Non-Uniform Tiling Arrangements?(https://commons.wikimedia.org/wiki/Category:Non-uniform_tilings_by_regular_polygons)
But to understand these (beautiful) pictures goes far beyond my knowledge, and I am still not sure if the sample space of the Honeycomb Conjecture contains finite or infinite packing arrangements.
Can someone please help me understand this?