What are these strange dice? (Part 2) - Numberphile

Featuring Henry Segerman. See Part 1 at • What are these strange...
More links & stuff in full description below ↓↓↓
Henry and Robert Fathauer designed these "TheDiceLab" dice - more info at: thedicelab.com
More dice videos: bit.ly/Dice_Videos
More Numberphile videos with Henry Segerman: bit.ly/Segerman_Videos
Humble Pi by Matt Parker: amzn.to/4a2FIhz
The idea to make two/three/four sided dice was suggested by Jimmy Kerl.
Patreon: / numberphile
Numberphile is supported by Jane Street. Learn more about them (and exciting career opportunities) at: bit.ly/numberphile-janestreet
We're also supported by the Simons Laufer Mathematical Sciences Institute (formerly MSRI): bit.ly/MSRINumberphile
Our thanks also to the Simons Foundation: www.simonsfoundation.org
NUMBERPHILE
Website: www.numberphile.com/
Numberphile on Facebook: / numberphile
Numberphile tweets: / numberphile
Subscribe: bit.ly/Numberphile_Sub
Videos by Brady Haran
Numberphile T-Shirts and Merch: teespring.com/stores/numberphile
Brady's videos subreddit: / bradyharan
Brady's latest videos across all channels: www.bradyharanblog.com/
Sign up for (occasional) emails: eepurl.com/YdjL9

[@numberphile]

See Part 1: kzbin.info/www/bejne/aZLOc4CJf5hqi80 Buy the dice at: www.mathartfun.com/DiceLabDice.html

[@Twitchi]

Man, that was a great looking book, If only I knew where to get it from...

[@mayukhpurkayastha2649]

W?-!at is this math India 🇮🇳🇮🇳

[@arikwolf3777]

So when are you going to make tetrahedral d2's in the style of "The Royal Game of Ur"?

[@Ansarkhan01780]

Support

[@Ansarkhan01780]

500 hundred subscriber

[@lowellrindler9454]

“As a pure mathematician it is my inalienable right to do pointless things because they’re fun and cause you can.” I’d like to be a mathematician and this truly fills me with the utmost hope and determination. I want to do pointless things because they’re fun and I can.

[@tiagom1665]

+1

[@jyrinx]

Some mathematicians even get a bit sad if it turns out something is useful

[@whatthefunction9140]

I want a t shirt with this quote

[@mrphlip]

​@@jyrinxThere was a video on another maths channel... I forget what the topic was, I think it might have been an Another Roof video? Anyway, as an aside in the video they say "unfortunately, this does have applications", and that's a mindset that has stuck with me for a while...

[@Math_Rap_and_GOP_Politics]

Paul Erdös felt this way too.

[@costa_marco]

I am always impressed on how Brady asks the best questions at the right time. I was just thinking about the fairness/independence of the results.

[@AaronOfMpls]

Indeed! And yah, years of experience, I imagine; he's been making Numberphile for more than long enough to _think_ like a mathematician sometimes. And he trained as a journalist well before that.

[@rustyrapp5673]

It’s almost as if these videos are produced in advance of filming…. Oh wait

[@unvergebeneid]

The d4 are actually amazing! Not only are they nerdy and beautiful, they also roll much better than tetrahedral d4! Like, the bane of everyone having to roll d4s is that the just won't effing roll!

[@icerifle7400]

"That's pointless!" "Thankyou" 😂 gotta love a mathematician

[@soranuareane]

I'd love to see the 2-3-4 dice poured in a clear resin with the inner symmetry (or just the cube) visible, like in Brady's visualization.

[@michaelslee4336]

Their prices are crazy and I mean that in a good way. All of that design effort, outlay for tooling etc and they are not that expensive for what they are. Kudos.

[@LightPink]

They are all solid colored tho

[@keithtwombley]

when those dice of constant width are available, you'll sell a bajillion.

[@HomeOnTheEdge]

I love those constant-width d4s! I can't wait to buy a set

[@shobhitkaul8076]

Love that you pick Matt's book

[@plackt]

That first one would be great for my minigun damage rolls! Xd4 is too much, and a “crit” 4 being more likely to roll a 1 just seems like good balance to me.

[@brikilian7834]

It isn't more likely to roll anything. It's that certain combinations are more or less likely. There is a 1 in 4 chance of each number on a d4. But if the f4 turns up as a 4, the d2 was more likely to be a 1 than a 2. So, it only works the way you wish if your game requires that a crucial is rolled on a smaller die.

[@riuphane]

I am 100% adding these to my Christmas and birthday lists

[@DDvargas123]

I remember learning about C2 continuity from The Continuity of Splines video would be cool to see a numberphile approach to the explanation of the different levels of continuity

[@agargamer6759]

Haha, I was wondering why that term was familiar! Been a while since I watched that video

[@PhantomKING113]

Nice c: Wouldn't it also be cool if the numbers on those constant width d4 were on the faces? They're see-through, so you could read them.

[@carnsoaks1]

Those clear dice need to come in 6 inch 'diameter' or larger. Super cool art pieces.

[@alveolate]

they're so insanely smooooth that they look pretty just all on their own! the fun bonus that they spin really well is also kinda cool. and oh, they do function as dice xD

[@HalfpennyTerwilliger]

I'm a tabletop RPG player and a math nerd. I did buy several of those and use them regularly while playing. The skewed d6 is my favorite.

[@Valyn]

The place people buy strange and unique dice are at gaming conventions. Places like PAX. Everyone wants those own unique dice. Sometimes unique sets for each character they make. The dice there go for up to 100's of dollars per set. Some made from metal, wood or have custom designs embedded within them. These these unique shapes could do well. Not sure about the 'recast' dice though, just because there are modification to each die, like rerolling 1s that would be awkward.

[@rngwrldngnr]

Similarly while I really like the schlerman dice, I think needing to keep them separate for reliable results is probably a deal breaker (though distinctive coloring and size would help).

[@unvergebeneid]

Another place where people buy these dice is the internet 😄

[@wobblysauce]

And only useful as 2d6, of course, you could roll multiple pairs.

[@GasparLewis]

Pleasantly surprised at the Grand Illusions mention, surprised no talk of Louis Zocchi or any of his famous exhortations on dice-making. In particular, his spiel is pretty zealous about NOT putting dice through a tumbler and preferring to live with the blemish left by clipping and sanding down from a casting sprue instead. It's compelling enough to watch in full and at least makes a decent case for it, and he goes into fascinating specifics about the manufacturing process.

[@andrewmartin3671]

Yes! Grand Illusions collab when? ;)

[@BooBaddyBig]

Those constant width dice are beautiful.

[@mesaprime4368]

"As a mathematician it is my inalienable right to do pointless things because it's fun" spoke straight to my soul

[@cadekachelmeier7251]

There was a Kickstarter a while back for a D Ultimate die. It's like your 2-3-4 die, but it has 120 faces and works as a 4, 6, 8, 10, 12, and 20 sided die. They're supposed to finally be shipping them out in a few months.

[@HenryCrun2]

You should do a video just on this 'C' concept of smoothness. Henry did a good quick job but I'm interested in learning more. EDIT: also I want to buy the constant width dice.

[@daemanuhr]

I don't see the C2-smooth d4 on the website yet. He did say it was a sneak peak, but I wonder how long until they're available. In the meantime, I noticed they have rhombic dodecahedra d12, which is one of my favorite shapes (it tiles 3d space, similar to how a hexagon tiles 2d space). I think those, as well as the recast 2d6 and Sicherman 2d6, will make a nice gift set for a math nerd.

[@alamos52]

A huge family of TTRPGs - "Powered by the Apocalypse" games - use 2d6 as the core resolution system. Rolling up to a PbtA game with those recast 2d6 would be a seriously sexy move.

[@Rubrickety]

That alternative d3 has an obvious advantage that would have jumped out to a more mercenary soul: there are three big blank rectangles for ad space. (Thinking about it, the inability to easily produce branded dice has probably vexed game manufacturers for ages.)

[@urgay1992]

Mercantile

[@phiefer3]

These are neat, however for the first set of dice in this video I can't help but say: d12 already exist and work as a stand in for a d4, d3 and d2.

[@Julio974]

Have you explored the idea of 4-dimensional dice? The values (D5, D8, D16, D24, D120, D600) for regular dice are easy to find, but a deeper look into the specifics and geometry and maybe simulation could be interesting

[@kiro9291]

that first one is so clever the constant-width dice look so fun

[@Anonymous-df8it]

For the constant-width d4, you could just take a regular tetrahedron and consider the spheres formed by having each vertex as the center and the others being points on each sphere, which, combined, would divide 3d space into 16 (unequal) sections. The section containing the tetrahedron would then be of constant width, and the edges and corners would be identical to each other (so they'd be equally affected by the tumbling process), analogous to the construction of the Reuleaux triangle; the proof is left as an exercise to the reader

[@Caesarr7]

The constant width d4 might genuinely be a better design than the sharp (dangerous!) normal design

[@mitigatedrisk4264]

And they look delicious

[@AaronOfMpls]

@@mitigatedrisk4264 Indeed, I could see someone making these in rock-candy or jellybean form. 😋

[@geoffstrickler]

The recast 2d6 has the same probability of sums, but “doubles” are drastically reduced as it’s only possible to get double 1, 2, or 3.

[@r3hawk]

I was thinking the same thing. Would be interesting to play Monopoly or Parcheesi with them.

[@joshuagardner4095]

I could see a game designed around the recast 2d6. Something where the total determines your movement and then the d3 determines some other outcome (like fight/flight/freeze)

[@joshuagardner4095]

Or you control 3 game pieces/characters and the d3 determines which actually moves.

[@HunterHouse43]

Those first dice were way to much for my morning fog brain

[@babel_]

Looking forward to the d4 of Constant Width! Be interesting to explore dice with more "faces" that retain constant width (and are C2 smooth), I haven't seen any 3D shapes outside the Reuleaux/Meissner tetrahedron, and it seems like it'd be weird if that was the only useful one given how 2D permits so many.

[@Tumbolisu]

You can take any 2D shape of constant width and just spin it around to generate a 3D one, meaning there are infinitely many 3D shapes of constant width. However, these are not very interesting.

[@Deutschebahn]

Those d4 are some of the most beautiful objects ever to watch

[@yarone5960]

Huge miss with the transparent d4 - should have put a mirrored number on the bottom

[@JAzzWoods-ik4vv]

>"there aren't enough... dimentions or something" lol

[@healplease]

shame this video is unlisted, this one features Matt Parker (indirectly) and dices in this video are so much fun!

[@numberphile]

It will be listed soon - I always make "part 2" videos unlisted at first so that people are more likely to see Part 1 first!

[@Anonymous-df8it]

@@numberphile That's clever! :)

[@TheGsoffer]

He engineered the new dice to the requerments! The is just beautiful.

[@EPMTUNES]

I like the engineering connections in this one!

[@triangledefinition]

I made some dice that give me two random numbers between 1 and 8 and then I flip a coin to choose the higher or lower, and I use it to look for random book pages to make art from. So I might make a number between 12 and 87 or if I added them together between 3 and 15, but they don't hit every number combination. For example I can't make 47 or 74 with my original set, but I did make two complementary sets that together can do every number between 12 and 87. They're a bit like those 'properly pointless' dice that make up the two 6s.

[@fierydino9402]

The provided animations are definitely brilliant!!

[@aner_bda]

I like the d2/3/4 dice. Beats what I usually do and that's rolling a d6 and saying 1 or 4 is 1, 2 or 5 is 2, 3 or 6 is 3 for d3 or odd is 1, even is 2 for d2.

[@retraceyourvods]

There are enough ttrpgs that use just 2d6 to justify the 2d6 recast being a purchase you make

[@bobrong9645]

They are fun, but if your goal is playing, you're much better with 2 d6s, if only because you can easily use them as 1d6 if needed. Plus you probably already have plenty of them already. Heck, people who don't know what ttrpgs are probably already have plenty of them. You could argue that the d3 could also be useful, but then again, we all know how to make a d3 out of a d6. Of course, I'm not saying that to prevent you to buy them if you like them, just don't try to rationalize your purchase that way.

[@kyzer422]

@@bobrong9645 That's true, but I bet you could design an rpg to specifically use these, like maybe the dodecahedron represents the damage you deal or your "hit location" or something.

[@HalfpennyTerwilliger]

I actually have those dice in my dice pouch and use them everytime 2d6 comes up (greatsword damage in D&D for exemple).

[@TristanFrodelius]

I was under the impression that the edges of a 3D shape of constant width _couldn't_ be the same. I'd always heard so. Possibly in a Numberphile video even.

[@Methylshift]

Great video, love the dice vids

[@CptGallant]

I am a math nerd and a gamer, but it was explicitly the math nerdiness that made me buy some of these dice.

[@HailtotheKiin]

9:24 yay stand up maths reference! That book is great, maths as a baseline but honestly just a lot of interesting history

[@Arexsis]

Surpirsed you made this unlisted

[@twixerclawford]

All the part 2s start out unlisted

[@rngwrldngnr]

I'd have expected it to be over on numberphile2

[@GreenAppelPie]

That’s stupidly simple yet mega genius at the same time.

[@SeanTBarrett]

You can show the breakdown of the d3 + "d12" on the original 6x6 breakdown; group each row of 6 into two 3x1 blocks, and each 3x1 block is a fixed value on the "d12" plus the d3.

[@matthewludivico1714]

The next level is perhaps to make the three answers received per roll actually describe a mathematical relationship?

[@robo0428]

Woah woah woah, you really saved the best for last. I was (almost) bored up until them dice of constant width at the end. Shut up and take my money. Immediately.

[@liliwheeler2204]

To me the more intuitive way to determine what numbers need to be on the dodecahedron for the Recast 2d6 is to use the chart at 5:11 -Divide it into quadrants -The bottom right and upper left quadrants are the same -Imagine picking up the bottom half and overlapping it with the top half, so the matching quadrants are overlapping/next to each other -Note that every column is a set of three numbers {n, n+1, n+2} in that order, so the {1, 2, 3} dice fits perfectly with these rows -End result: 1, 2, 3, 4 (×2), 5 (×2), 6 (×2), 7, 8, 9

[@lucas.cardoso]

The easter egg in this video is a glimpse in which we can see how acquainted Henry is with poltergeists.

[@yomifishery]

Those d4s are the most delicious-looking dice I've ever seen.

[@LucaIlarioCarbonini]

Anyone selling a Gömböc with a "20" on top of it? Asking for a friend...

[@Lemau]

I *SO* want the Rouleaux dice! 🤩

[@glom9995]

Statistically better off staying put on a campsite or walking around when a search party is looking for you? Monty Hall problem?

[@WAMTAT]

These Dice are awesome

[@brunoramey50]

By sheer luck, you choose THAT book !

[@NikolajLepka]

I love Henry's work. It's always so wild and wacky and interesting in all the best posible ways. If only the shipping cost of the dice to europe wasn't so astronomical

[@loreleihillard5078]

I'm curious if theres a permutation of dice that has the same probability table as DnD stat rolls (i.e. 4d6 take away the lowest value). Might try to figure it out but if someone else wants to, please do

[@johnydl]

Can you get all of your dice on mathsgear in the UK too, I've had real trouble buying your d48, letter dice and optidice

[@RSVTuono]

"That's pointless!" "Oh, of course!"

[@Impatient_Ape]

For the math behind the Sicherman dice, Google "cyclotomic polynomials".

[@TiagoTiagoT]

For the constant width 4 sided dice, wouldn't it make sense to take advantage of the flatter "corners" to lay the numbers straight at the top instead of having to have them surrounding the corner in multiple copies?

[@henryseg]

I think we felt that it was better to stick to the "traditional" d4 numbering. Although the "corners" are flatter, they're not as flat as where we put the numbers, so the cuts would have to be deeper, making the plastic weaker and maybe messing with the smooth rolling a bit more.

[@jackee-is-silent2938]

There's also the weighted average 6-sized die, used in various paper wargames over the years back in the 1960's and later. It was 2 - 3 - 3 - 4 - 4 - 5. Could use regular dice and just read the 1 as a 3 and the 6 as a 4.

[@pieter-jan26]

Beautiful!

[@unvergebeneid]

Haha, after all the weird solutions to nonexistent problems he's heard, Brady is almost outraged at the 2d6 thing 😂 Actually, it's the first one where I was like "oh wow, that's actually ingenious!" 😅

[@frankharr9466]

Fascinating.

[@evertoaster]

I have ordered a set. :)

[@PoweredMinecart]

It would be cool to make a 2 and 18 sided die that act like 2x 6 sided dies. Or even better, a single 36 sided die.

[@paulfoss5385]

A 36 sided die would be neat, however 36 is not a factor of 48 (the order (size) of the full symmetry group of the cube including reflections) or 120 (the order of the symmetry group of the icosahedron), which would preclude anything nice looking. It would have to be a double cone type shape like the d10, but with 18 sides on each cone.

[@HeavyMetalMouse]

For the d2/d18, we can get {1, 4} + {1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 8} as the numberings. The d36 would be fairly straightforward, as you just have a single die, with known desired frequencies. To get really weird, for completeness, a d4/d9 solution could be {1, 2, 4, 5} + {1, 2, 3, 3, 4, 5, 5, 6, 7}

[@gfrank98]

You forgot the set of a d4 and a d9!

[@Pystro]

@@gfrank98 The only fair d9 is an 18-sided trapezohedron. If you used a dice of that shape, why not have that d18 and a d2. And it's also kind of close to a 36-sided bipyramid.

[@paulfoss5385]

@@Pystro Yes, though technically there would also be the nonagonal bipyramid, and family of skew in between shapes would also be acted upon by D9. And the dihedral groups are not the only family of subgroups of SO(3) which can have arbitrary factors, there's also the cyclic groups, which act on a ludicrous family of dice including the dice lab's sublimely unnecessary thirteen sided "roll a card" die.

[@mariegrover1409]

The recast 2d6 are my favorite!

[@LeoStaley]

You could theoretically do this with a D60, and replicate a d2. D3, d6, d10, d12, d20, and d30, but if would have to be huge to label it correctly

[@gcewing]

Those last ones are for making your game last longer by keeping you in suspense while they take forever to stop spinning.

[@MmmVomit]

I kinda want the recast 2d6 to use in PbtA games (table top role playing games where all the rolls are 2d6).

[@davidgillies620]

The mathematical justification for Sicherman dice in Wikipedia uses cyclotomic polynomials and generating functions to derive the result (I know 'cos I wrote it). A similar technique applies to other non-standard dice.

[@marksteers3424]

How do you account for the "weight" of the numbers - either if you add to the surface or conversely etch it - the number 4 requires more ink (or more hole) than the number 1?

[@anophelesnow3957]

On their site they have 'Optidice' - "dice optimized for fairness by using numerals that are physically balanced ", I do not know the bias regular numerals make.

[@AlyxGlide]

bringing chaos to the tabletop shops

[@TheZotmeister]

[raises hand] I bought the "recast 2d6" years ago.

[@Z_Inspector]

"I know a few math nerds" Sir, that's what the channel is about

[@impendio]

As a tabletop player, those are some nice ass dice.

[@Jack42494]

If any game is absolutely crying out for strange 2d6 dice, Troika definitely is! The recast 2d6 would be so fun to use with it!

[@jansenart0]

Just call the whole set "Chaos Dice"

[@AleksandrMotsjonov]

Love @standupmaths product placement ha ha!

[@Max_Flashheart]

But are they "Lucky Dice"? - In our D&D group we have special damage dice. They were giant brass D6 dice and they were heavy enough to do real harm if you dropped them on your foot.

[@KasranFox]

i really want to invent some kind of game you can play with the d4s of constant width where you roll them into each other like marbles

[@JigmeDatse]

The $C^2$ smooth d4 looks like it's right practical. I'm not sure if I really appreciate it, or find it to violates the, "what are these weird dice that are useless" property in ways that I find disturbing. I want to get some though.

[@Kufunninapuh]

The D4 just went from me least favourite to just favourite.

[@maxwchase]

Those "alternative 2d6" sets have me imagining what they would do to the probability distributions of landing on Monopoly squares. (Best to subject Monopoly to combinatorial analysis, rather than play it.)

[@notayoutubehandle]

The recast 2d6 are cool, but you can't use them to play Monopoly - you will never roll a double to get out of jail or get a free turn

[@storyspren]

I think "recast" works because it's like you cast different shapes in the role of two cubes, like how when an actor gets switched out for someone else the role is recast :D

[@anthonywestbrook2155]

Easy example of how the recast 2D6 could be useful. Lets say you're designing an expansion for Settlers of Catan. The core game is based on the distribution of 2D6. You want to add an effect that has three outcomes. You could add a third die, or you could use the recasts. Obviously that means you'd never have effect one happen when the total is 11 or 12, but that's unlikely to be a problem.

[@fisadev]

I need these die so much!! Sadly I'm from Argentina, so the shipping cost and chances of it getting stopped by customs are pretty high :(

[@gilmoses3777]

Beautiful. For the last, constant-width dice, why not engrave the numbers on top only once, rather than "triplicating" them around?

[@ollllj]

constant-width wonkel-engine-dice are nice for that extra utility. great caustics but only c2-tubler it, and never put it in a c3+ tumbler! 3-sided dice (that are not pencil/prism-shaped) alone sounds useful. the rest is just clever splitting of cube corners. of course those are dependent. surely, you can convolute a lot of dice-pairs wiedly, but that D3+D12 ==2d6 is just unnecessary, and the more you convolute anything, the faster you approach a normal-distribution anyways.

[@pn4960]

I feel like the dice of constant width are very smart, because they are harder to control. With the regular dice, you can train to make them fall more often on the face that you want. I think casinos might be interested by this design maybe 🤔

[@noahnaugler7611]

I may just so happen to be one of the few who fits in the categories of math nerd and tabletop gamer both. I am definately looking at the constant-width d4 I particularly am seeking to collect every isohedral shape in dice form (excluding the infinite families). Currently, among all such figures with opposing parallel faces, I am only missing the Triakis Icosahedron. Possibility of production and retail?

[@charllandsberg]

"Who is the market?" I AM THE MARKET 🤣🤣🤣 I love weird dice.

[@darbyl3872]

Why not roll a normal d12, and partition the possible results into 2s, 3, and 4s?