Sphere packing Puns

So I came across this post which shows a dramatic difference in the height of snowflakes in a cone based on orientation. I understand how this works with the height-volume relationship. However it got me thinking about packing. (I know next to nothing about packing)

If, instead of snowflakes, the vessel contained styrofoam balls, could there be a noticeable difference in height (controlling for the volume) based on packing efficiency alone? The differences could be a result of packing at the tip of a cone vs a cylinder or the tip of a cone vs the base of a cone.

Could the method of filling affect this? Eg pouring them in at random and seeing variations in height after shaking to rearrange.

Are there any interesting examples of this kind of phenomena?

I was doing the planning for a croquembouche, and that got me to thinking about the maximum number of cream puffs one could fit inside the (traditionally cone-shaped) dessert, which I thought I could model as spheres packed inside a cone. I would expect this amount to vary based on the cone's aperture, but I'm not sure, and my search for an answer to this question only turned up papers on packing cones inside of other things. Can anyone answer my question/point me in the right direction?

As a chemist, I'm could probably figure this out, but that math portion of my brain is a bit rusty, so here I am.

I make mechanical foams out of polymer. To make this foam, you fill a mold with dissolvable, round beads. The beads have to touch, or else there is no way for the solvent to reach them. The polymer is injected into the mold, and it fills in the voids around the beads. After the polymer cures, the beads are dissolved out and you get a really nice foam with uniform pore size. My other way of making a foam with this polymer, is modifying the chemistry, such that a gas is generated during curing, but this gas expansion method isn't the most controllable, and the resulting foam isn't as nice looking.

The dissolvable round beads are pretty close together in the mold. I'm pretty sure I'm getting a packing density of 0.625 to 0.641, based on some density tests I've done. It would be a close random packing, inside the mold. I want to change the packing density to something more like 0.55, maybe even lower.

Could this be done by using different sized dissolvable beads? I want the beads to still touch, but I want poor packing and larger void spaces to fill with polymer. Currently, I use a 2 mm bead.

I am considering sphere packing (recreationally), but I'm making my spheres out of Play Doh (figuratively). My problem is like this: pack n spheres into a container maximally. Take each of those spheres and cut the Play Doh into m even parts and form them into spheres again. Under what conditions will the set of mn spheres, each with (1/m)th the volume of the original spheres, fit back into the container?

For a base case, consider a unit sphere in a unit box. If the volume is reduced by a factor of 2, 3, ..., 7, they won't fit back in, but when reduced to 1/8th volume, those 8 spheres will fit in the unit box in a 2x2x2 grid. Same with any power of 3, but I don't know about 9, 10, ... 26.

What keywords can I search to read about a problem like this? Any good sources to look at? I don't have very good intuition for physical problems like this, so this might even just be a dumb thing to think about.

Side note: I thought of this because last night I chopped up a container of mushrooms, and the chopped mushrooms barely fit back into the container that held the 7-8 whole ones.

I just came up with a the following problem:

Consider a 10 by 10 square. What is the minimum number of discs (of radius 1) that you can fit into the square, such that no further disks would fit?

Of course this problem can be asked for other shapes and dimensions. Do you guys have any approaches?

You can fill 100% of space with packed spheres.

First note that a sphere and a cube are topologically equivalent. So just pack space with cubes!

It has no empty space left and because the cubes are spheres you just packed all of space with spheres!

Asking for a friend.

Buckminster Fuller was incredibly close to figuring this out, and laid the groundwork for the energy dynamics of the geometry of this theory.

One of Fuller’s many significant contributions is his epic tome, Synergetics: Explorations in the Geometry of Thinking . This exhaustive study of the patterns and structures that are inherent in energy dynamics is remarkable in both scope and depth of exploration. In Synergetics, Fuller expounds on key concepts that have informed the theory of Unified Physics in fundamental ways, including among others:

• the Triangle as the most basic unit of all structure, and the Tetrahedron as the most basic system of energy dynamics;

• the Isotropic Vector Matrix as an underlying cosmic energy lattice and ultimate “source” of all localized pattern and structure;

• the Vector Equilibrium as the conceptual “zerophase” of all energetic systems;

• the “Closest ­packing of Spheres” that inherently constructs these energetic forms

*and the dynamic “jitterbug” pulsation of the Vector Equilibrium that creates all primary (platonic) forms and spiral vortex flow dynamics.

>“The vector equilibrium is the true zero reference of the energetic mathematics… the zerophase of conceptual integrity inherent in the positive and negative asymmetries that propagate the differentials of consciousness.”

• Buckminster Fuller, Synergetics

>“Omnitriangulated geodesic spheres consisting exclusively of three­ way interacting great circles are realizations of gravitational field patterns...

The gravitational field will ultimately be disclosed as ultra high­ frequency tensegrity geodesic spheres. Nothing else.” - Buckminster Fuller

From Quantum Gravity and the Holographic Mass

>It then follows that the Schwarzschild solution to Einstein’s field equations could have been developed in the late 19th Century by computation of tiling Planck quantities independent of spacetime curvature and singularities, near the time when Max Planck in 1899 derived his units.

My background is in programming, not math. But I am building a house using rammed earth and I think this problem applies. How do I find the right mixture of rock/sand to fill a space with the least amount of empty space.

Let's say I have 9 different sizes of aggregate starting around 19mm and halving all the way down to 0.075mm. It seems like there would be a way to create a computer model to calculate the correct ratio of each sized aggregate to achieve the highest possible density given an "x" sized cube (30cm).

I know there are practical tests that can be done to get me close but I have been rattling this problem around in my head for months now and I can't sleep so I'm reaching out to the internet.

If you have any papers or books I could start with it would be much appreciated. Thanks

Hello guys, I've stumbled over this problem in social media. It is a rectangle of length 100 cm, height 150 cm and width 80 cm in which you have to fit spheres with a diamenter of 8 cm. How many fit into the rectangle?

I tried solving this by dividing each side by the diameter of the spheres and multiplying the results, but I doubt that's the most efficient way they can be packed.

Hi, my class recently did an investigation where we put marbles in a cylinder half filled with water and saw the correlation between the increase of water level and the number of marbles added.

Now we are trying to work out the most effective way to pack these marbles to maximize how many marbles we can fit in the cylinder. We’re stuck and I would love some help to try to solve it.

We were also discussing the void between each marble and the average percentage of the whole cylinder which will be void compared to marbles once filled with marbles.

Average marble diameter= 15.890621 mm Cylinder height= 197 mm Cylinder diameter= 42mm

So I barely know anything about math, but when it gets slow at work and I get bored, I start to wonder about weird little puzzles and stuff, and here's the one that I'm currently stuck on:

First I wondered how many spheres could touch a center sphere at one time if they were all the same size. So after fumbling around with some marbles and a hot glue gun I eventually discovered something that apparently has been common knowledge for hundreds of years: the answer is 12.

But look closely. This solution ends up leaving a small gap between all of the exterior spheres - in other words they're not touching one another. So basically if you increased the sie of the exterior spheres slightly, then they would all be packed as densely at possible, with each sphere coming into contact with all it's neighbors in a sort of triangular grid, you know what I mean?

Anyways, my new question is a little bit hard for me to word, but hopefully you'll all understand and maybe be able to help me out here:

Basically imagine that I pick a central sphere with a diameter of 1. Then, I pick a specific number for how many spheres I want to pack around it so that they're all touching (for instance, 4 spheres surrounding it). And the question is: what size do those exterior spheres need to be so that they all packed as densely together as possible with no gaps. Okay, hopefully that makes some amount of sense.

I went into the little Windows 3d Builder program and messed around, and here are some of the rough solutions I've found, which probably aren't very accurate, but its a start:

• 4 spheres: roughly 4.48
• 6 spheres: roughly 2.42

So anyways, I'm just looking for any help, advice, or solutions to this problem. Hopefully someone else finds this interesting, hah.

The sphere packing problem essentially asks, what is the most efficient way to arrange congruent balls as densely as possible, without overlap.

Recently, two papers were published on the sphere packing problem in higher dimensions, the first in dimension eight and the second in dimension 24.

The maths is beyond me, however the conclusion drawn from these two proofs was that in these higher dimensions, sphere packing is more efficient/dense than in just 3 dimensions. Any help as to why would be much appreciated.

I'm just starting to dip my toes into studdying this area and would like a little overview of what has been discovered.

How has this been generalised to other normed fields?

Are there any active areas of research?

Thank5

I came across this article while researching sphere packing. I decided to implement the algorithm on my own and produced this. The algorithm works by finding the point(s) furthest from other shapes and the boundary, generating circles there, and repeating. However, if the generated circles only take up a fixed proportion of the space available, there can be strange overlapping patterns (try setting circleScaling to 0.35 in my simulation). I want to figure out which proportions result in overlapping circles and which do not. Is there a name for this sort of packing? The blog post used "Apollonian packing", but this is usually reserved for a circular boundary.

This article here talks about how a proven optimal packing in dimensions 8 and 24 has been found. I had some questions that maybe you guys can help with -

Do the E8 and Leech lattice packings leave no wasted space? In 3 dimensions, rectangular shapes can be maximally packed with no wasted space, but spheres end up wasting space. If the E8 and Leech packing don't waste space, then what does that imply about the overall shape of those dimensions? Seems to me that since rectangular shapes can be perfectly packed in 3D, then 3 dimensional space is rectangular. Are the 8th and 24th dimensions "spherical" in nature?

Also, can someone ELI5 modular forms? I have an engineering background, but am certainly not a mathematician.