Why Do Bees Make Rhombic Dodecahedrons? 

I'm afraid you're wrong Matt, bees aren't mathematicians, they're engineers. There might be more efficient shapes but some poor bee-geneer has to make the thing. The bee project manager probably got a bulk discount on off the shelf regular rhombii rather than ordering bespoke wonky hexagons and squares. Think of the extra paperwork!!

"hexagons are really a good shape" is the Parker's Bestagon

You forget the importance of symmetry. I think the reason bees couldn't build truncated octahedrons is because it would require them to choose a direction (for back-to-back assembly), and make sure every builder follows it. While dodecahedrons will always merge perfectly together, no matter how many starting points there are and wherever they are. They are symmetrical. So dodecahedrons is definitely the best shape for distributed workforce.

The "freestyle comb" honestly looks like an art piece to disrupt wind turbulence/resonance.

Weirdly, I really do complain a lot less when I'm not being actively stung by bees!

If it isn't my favorite 40 year old Australian recreational mathematician, author, comedian, RU-vid personality and science communicator!

"Alex complains a lot less if he's not actively stung by bees" - ugh, such a needy camera man!

I wonder what those bees must think of someone just casually taking the roof off their home, pulling out an entire interior wall, and waving it around. They seem remarkably chill about the whole thing.

HEXAGONS ARE BESTAGONS!!! So glad he got that CPG Grey reference in there!

Although it doesn't use the optimal amount of wax for the space, perhaps it is still the most efficient shape for them to build, because of the "hardware" they have. It is simpler in it's symmetry, and tiles more naturally than the other solution you presented, so I can imagine it would be simpler, for example, to program a machine to build the rhombic dodecahedron than to program it to build a truncated octahedron...

The rhombic dodecahedron seems to be better for stability, as it shares the bottom surface with THREE cells on the other side. The truncated octahedron only shares with TWO cells.

A wise man once said: "Hexagons are the bestagons."

Optimal over what feature space? It could be that the rhombic dodecahedron, while using more wax, provides better structural support, has better thermal properties, etc.

0:12 missed opportunity to say hexagons are the bestagons. 10 seconds later: Why did I ever doubt you Matt?

Sometimes I see RU-vidrs I haven't seen before, and they are smart, talented, charismatic people. Maddie is the first one in recent memory who made me think "Wow, this is indeed a TV presenter". The level of personalism is fascinating

"Hexagons are the bestagons" - CGP Grey

When you inserted the rhombic dodecahedron into the hexagon, I was stunned to silence! What a beautiful fit for a great shape. Thanks for making models, I'm a visual learner, and your models are helpful.

Learn the beauty of the Truncated Octahedron space filling shape! Since bees make round 2D combs that settle into a hexagonal lattice due to wax pressures distributing, they're taking a 2D optimization and extruding it into 3D space. If a 4D bee were to take 3D spheres and pack them together -- as they do with hexagons -- then the wax would settle into 3D Truncated Octahedrons extruded into 4D space.

The fly of the Bumblebee by Rimsky-Korsakov was such a nice touch. Once again I'm super impressed by the production quality of this videos. Really, really nice.

I hereby request a collab with CGP Gray to discuss 3D best-ahedras.

The CGPGrey reference was really well done! Starting off with a misdirect and then just as this CGP viewer was thinking Matt really should have referenced him, sneaking it in the after all. Loved it!

I think the important part about the end capping isn't the efficiency of the capping, but how it fits together. The shape bees use really obviously slots together with the opposite side of they're just offset. More complicated end cappings may save wax on an individual side, but if it's not the most efficient solution for capping both sides at the same time it's still advantageous to use the rhombus endcaps.

Figures it's something as simple as that. Hexagons just have a tendency to pop out when designing parts and more often than not when your trying to connect all the components that make up a given part. They are a fairly easy shape to produce on accident.

The ending was amazing. It threw me all the way back to 2012 when ViHart made a video series on plant geometry (which I will of course rewatch now) It's fascinating how intricate mathematical patterns arise from very simple instructions.

As a bee-keeper I loved this video, but you should have used foundationless frames if you're going to look at their cells in this level of precision. Glad you picked up that they're tubes more than hexagons, and rounded bases more than polygonal ones - but bees also build cells in multiple sizes, something that foundation prints try to discourage. If you look at mixed-cell comb, I think you'll find the structure makes more sense. You kind of got there looking at the edges.

I find it more fascinating that they create these shapes by iteratively reshaping the wax! Doing geometry the way we do isn't necessarily the most "natural" way one may say.

I'm glad to see some appreciation for the cinematic masterpiece that is "The Swarm".

One-upping (a pun on going to a higher dimension) "hexagons are the bestagons" is the most on-brand thing for the person who wrote the book on the fourth dimension

11:28 Could you show us how the hexagonal prism into truncated octahedron fit together? What would be the offset arrangement of the two sides of the honeycomb? Edit: The arrangement is that the hexagons of one side are centred over the middle of an edge on the other side. The peak edge where the two squashed hexagons of truncated octahedron meet is exactly where a wall is built up on the other side. The end to the prism is oriented the same for all the cells so, if you look at only the small rhombi, you see many parallel straight chains running across the pattern.

This one is going to live in my warmed heart for a while now. I adore this

Love to get your take on the fact that the cells are built with a slight tilt upwards towards the open end, between 9deg to 14deg on average.

I'm surprised you skipped over a very important detail. The cells of the comb that they build are not horizontal, they are sloped upwards at between 5 and 7 degrees. Your model presupposed that the cells are aligned parallel to each other on opposite sides of the comb, but that is not the case. Also, another minor detail, you implied that they build upwards, starting in lower cells and moving on to higher cells. That is also incorrect. They start at the top and build downwards.

I was thinking about it - and I suspect there's a simpler explanation. Fundamentally the cells are a stack of cylinders with hemispherical end caps as a circle has a minimum circumference per area and a sphere has a minimum surface area per volume. However, this stack leaves a lot of wasted volume between the cylinders, so the walls are pushed out until they meet - creating the existing cell shape.

would bees accept a hotel with squared cells? (kind of a parker-bee-hive)

12:02 it might be possible that another reason for them not using a more complicated packing shape is that their edges have volume- having a face with 5 edges vs 3 edges like in the rddh would mean more wax would be needed for each edge, which might not pack as tightly.

There's nothing more awesome than watching standup maths in 360p

When talking about the best shape, you may want to consider that a bee only needs to know 1 shape (the rhombus) and no consideration of orientation to create the cavity (requiring only 1 type of bee) rather than many shapes in particular orientations (or many kinds of bees or some bee micro-manager). The shape may not be the most efficient, but it seems the most practical for simple automatons. Very fun video tho :)

I instantly knew what movie the clips were from, since I've watched one of the early episodes of Citation Needed!

Looking forward to the second channel video with the Bee-roll footage.

Maddie has such an infectious smile

The Swarm! From Michael Caine's "It's A Living" period! Made internet famous by the Technical Difficulties!

"The bees aren't doing maths; the maths is doing the bees"..... will quote

I only have 360p as an option. Might only be me.

Oh god, a clip from The Swarm right off the bat? Man after my own heart

17:15, Matt, you gotta start normalizing the volume in your videos mate. Your aside was so quiet, and then you come back in yelling. Idk if you use Final Cut, but I know at least if you use Premiere Pro, you can run a loudness report on the export screen and it will give you an impression of your maximum loudness. You can help a lot by adding a bit of compression, a limiter, or even just watching your video back one last time with the audio level visible to make sure you aren't peaking when you have your volume set to a place where the preceding most important parts of your explanation are actually audible.

Clearly Matt is a Latinophile. He goes with the Latin plural "rhombi" but not the Greek plural "dodecahedra."

Stuff like practical geometry is really fun. Polyhedra like those can be found also in crystals an crystal lattices, and the rhombic dodecahedron is also the Wigner-Seitz cell of the face centered cubic laatice, one of the two most dense arrangement of spheres in 3d (the other being the hexagonal lattice, hcp)

I like what you did with the patreon names. It looks like no one else has commented it. But I noticed! Was worth the effort Matt. :)

I always wondered what a "natural" way of filling 3D space would be.. and yes studied some foam experiments I did. As far I can tell.. there isn't a clean winner, it's always kind of chaotic contrary to 2d space (like on the surface of a fluid) where it will create hexagons. So I guess this means there are a lot of possiblities to fill 3D space, but almost as good, so chaotic shapes are created.

Just looking for information about Hexagons in hives for a school project and found this, thank you man

I was so not expecting that mention (and slight teasing dig) at CPGrey but MAAAAN in was glorious!

The regular dodecahedron is my favourite of all the dodecahedra, because the regular pentagon is a very elegant shape, I think. The only problem: The regular dodecahedron doesn’t tile Euclidean 3D-space, by itself.

As Matt and Maddie approach the hives at 4:48 the background music is “Flight of the Bumblebee” I wonder how many viewers caught that Easter Egg?

Would be interesting to slowly rotate the frame, or in increments to see how much they use gravity. Could you get a swirl?

The 120 degree angle occurs naturally because it minimizes the surface energy. The bees actually make circles but the the wax is a flowing solid which minimizes the overall energy when all intersections are 120 degrees. This is also how crystal grains grow is solids such as when metal is annealed.

I want to see an ultra-low-budget sci-fi film with those beekeeper suits for spacesuits. ;)

0:40 hexagons are the bestagons! CGP converted you to the religion as well. I hope he will watch this video!

I can't believe Matt missed the opportunity to reply to "You happy?" with "Absolutely buzzing!"

Bees are great! They're so fascinating; everything from their wiggle dances to communicate the location of flowers, to their honeycomb. I like to watch them at work sometimes when I chance upon flowers with busy bees around.

4:10 "I've worn a bee costume before, but it was different... I was literally dressed up as a bee ! " - Matt

11:28 Is the claim of more efficient filling just for rhombic dodecahedron *(without hexagonal prism)* vs truncated octahedron *(without hexagonal prism)* at the goal of *fully filling 3d space?* Or is that claim for *hexagonal cells* with rhombic dodecahedron shaped backing vs *hexagonal cells* with truncated octahedron shaped backing at the goal of *being built upon the two sides of a plane with all the cells open to be able to be accessed?* Edit: the efficiency boost is for the bee cells, but the area saved is only 0.35% of the mouth of the cell, it assumes the area of 0-thickness walls is a good measure, and losses could be introduced by 'crystal grain boundary'-like inefficiencies. 'crystal grain boundary'-like inefficiencies: When the bees start to make cells on side B, to get the truncated octahedron shape, they would start with a cell centred on an edge of the hexagonal grid of side A. But there are 3 possible phases. So, unless they had a plan of where to start building, if multiple starts are made on side B, when these growing honeycomb 'crystals' meet, there would be a 2/3 chance of misalignment.

I haven't believed in bee-made "hexagons" for many decades now. After thinking about it. They are merely circular / tubular cells, stacked like you would cut firewood logs. The negative volume of which. The walls separating the cells are the natural gaps between the circular chambers. If you thin those walls out they will eventually transform into more evenly thickness surfaces / walls, which at a distance resemble hexagons. Matt covers this thinning out or pressure from adjacent chambers. It's not that bees are clever architects but more about cells or soap-film like geometric efficiencies manifesting naturally. Equilibrium. Again, Matt mentions soap film after I got to that part.

this answers so many questions I had never looked into Thanks!

I was muttering soap bubbles under my breath for the whole video. Made me happy when you used it as an example at the end!

So the bees aren't trying to make hexagons per se, they're just trying to stuff the most holes into a limited space and that's just how it shakes out. Very interesting.

Me : *Grabs a bee* Me : Why are you making Rhombic Dodecahedrons? Wasp : *Visibly confused* Me : Your tricks won't work on me, fool.

Amazing video I can see the passion you have for the math in this Cant remember who it was that mentioned your channel, but I'm glad I subscribed to it

I love that math didn't come up with it. It was a goal and a process that ended up being what math can describe. The bee's goal is to get in there and cull out material while making the largest volume, and what comes out is what math can describe. Now I'm thinking of entropy and baby steps.

Hexagons are the bestagons!

Matt - you should get that small bit of wax structure set in a block of epoxy as a desk ornament👍👍👍

I'm not that into bees, but this video is wholesome. Thank you.

Facinating. I was reading a beginnner beekeeper's handbook, and didn't realize before the 20th century people had to kill the bees to get the honey. The whole hive. The removable frames made it possible to reuse! Crazy

For someone who likes maths and bees, and works at a factory that packages honey, this was interesting.

The clips from The Swarm killed me, Matt

Thank you, Matt, for this birthday present!!

Your videos are such good mathematical concept education. Thank you for teaching over my life!

I've been waiting for this video my whole life. I was the 66th like. I love you Matt. In a platonic way.

I love Bees me and my fiancee went to a bee farm the day we got together. over 4 years ago. when i proposed I actually used a beeswax candle charm that I ordered from the exact farm that we went to. and I was really corny and said "will you BEE my wife"

As soon as he said bestagon I subscribed, can't pass up someone who knows the universal truth

Imagine what shape a 4D hypebee would make their hypercomb. The hole would approximate a rhombic dodecahedron and I can't conceive what fantastic shape the back of the tube would be.

I’ve been a part of the “hexagons are the bestagons” cult for a good while; I am now happily a part of the “rhombic dodecahedrons are the bestahedrons” cult.

Hexagons are bestagons

13:25 - The proper term is ‘recovering’ programmer.

Delightful, as always!

Why is the video quality maxed out at 360p?

Of course they're suboptimal in 3D space they are only B's, if they solved it optimal they would be A's

I keep bees too! If you take some old dark comb and leave it to weather in the sun the wax cells will delaminate from each other and you'll be left with many individual cells of the shape he made from paper! They really are amazing!

Noice use of 'Flight of the Bumblebee'

Maddie is smart, sharp and charming. Just like you, Matt!

Really, it's a rhombic dodecahedrons on one end, hexagonal prism on the other... has anyone named that shape yet?

11:52 "This isn't even the best solution!" -- Strong Dave Gorman vibes there.

9:46 I think Matt has reached his lifelong peak for gratification

Come on Matt, you could have gone with "A-Level Knowledge" as well, for "Apiarist"

Hexagons are the bestagons

You can hear the offset phasing of the frequencies of Matt's voice reflecting off his paper sculpture at 18:25 as he pulls it away from his face.

a) love matts visual aids b) second video I’ve watched this week to reference The Swarm

"Hexagons are..." "Come on, say it!" "a really good shape" *angry viewer noices* "However, hexagons are bestagons." *happy viewer noices*

Hexagons are the bestagons! Except when you need to make a spherical object (such as a soccer ball), in which case you'd need to add some pentagons into the mix. But I don't think Matt would know anything about that. _Right?..._

This is delightfully wholesome

Lot of good bee-roll footage in this one, nice work!