How many 3D nets does a 4D hypercube have?

The cubes I am using are called "mathlink" and I just bought a huge quantity from Amazon (because Learning Resources didn't answer my emails).

The unfolding animation of the 'Dali cross' was kindly made by my Patreon supporter John Sawyer.

CORRECTIONS:
- I saw "288" at the end of the 8D number when it should be "228". The on-screen number is correct. I noticed too late to fix it!
- At 21:09 I say Diaz and O’Rourke found an unfolding of the Dali cross which tiles the plane. It’s actually a different 3D net they found which does this and the Dali is undetermined if it produces a tiling 2D net. (Thanks Dan L by email.)
- Let me know if you spot any more mistakes!

Filming and editing by Alex Genn-Bash
Maths graphics by Matt Parker
Music by Howard Carter
Yeah, I decided to replace the copyright-claimed Aerosmith.
Design by Simon Wright and Adam Robinson

MATT PARKER: Stand-up Mathematician

[jamesoh]

"You can be part of mathematical history!"
*1 day later*
"History is full. Go away"

[MichaelWilliams-owue]

The definition of a good fanbase: Literally the next day and every single tile has a submission on the website

[spaceshipable]

As a side-note, Minecraft turns out to be a really useful tool for playing with the tilings!

[zinsch]

Matt Parker "It's not going to be quick."
Narrator: It was going to be quick.

[CameronWinters]

“Unfolding the Hypercube” sounds like part 1 of a prog-rock concept album.

[s_gaming]

holy crap i dont know if its just me but that music that suddenly starts at 10:07 was loud it almost blew my entire house into orbit !!!

[bloodgain]

"Classic mathematician"
An engineer, a physicist and a mathematician spend the night in the same hotel.
At midnight, the engineer is awakened by the smell of smoke. He takes a step down the hall and sees a small fire. Thinking fast, he dumps his wastebasket, fills it with water, and puts out the flames. Satisfied, he goes back to bed.

Later on, the physicist is also awakened by the smell of smoke. When he investigates, he finds a second fire in the hall. He runs to the end of the hall and picks up a fire extinguisher. In the time it takes to do this, the fire grows in size, but he is able to do some quick mental calculations and find the ideal place to aim the fire extinguisher nozzle. He manages to put out the fire and returns the extinguisher to its support. Satisfied, he goes back to bed.

Finally, near dawn, the mathematician wakes up to the smell of smoke. He walks down the hall and - you guessed it - finds a fire. Looking up and down the hall, he finds the fire extinguisher in its holder. "Aha!", he says triumphantly, "there is a solution!" Satisfied, he goes back to bed.

[joesankar]

And less than 24 hours later, there are multiple submissions for every 4D net... I'm genuinely impressed, well done internet

[MariaVlasiou]

You could have a complete taskmaster series with standup mathematicians

[StormeeSkyes]

As a secondary school teacher I can only dream of owning that many multilink cubes!

[jotha]

18:07 There was actually a different third emotion in my mind: "Wait, the Parker tilling is actually a real tilling??!?" ;)

[Solitaan]

"It's not going to be quick"... yeah, later the same day and there's a submission for each one! Good job yall!

[TomE]

This video was quieter than the adverts by quite a margin

[atimholt]

The 120-cell is my favorite shape, so you got me curious about its nets. I googled “120-cell number of nets” and found a paper from 1998 called “The number of nets of the regular convex polytopes in dimension ≤ 4”. Notably, the authors are named F. Buekenhout and *M. Parker*. Thatʼs nothing to do with you, right?

EDIT: Hereʼs the abstract:
Classifying the nets (also called unfoldings or developments or patterns) of the regular convex polytopes under the isometry group of the polytope is equivalent to classifying the spanning trees of the facet-adjacency graph under its automorphism group. This is done for all such polytopes of dimension at most 4. © 1998 Elsevier Science B.V. All rights reserved

[brandonmack]

I love that, at the mathsgear link, it says that the nets can't be folded back into a 4D hypercube without highly specialized equipment.

[RedHair]

The fact that you can unfold a 4D cube into a 3D shape that perfectly tiles a 3D space, that can be unfolded into a 2D shape that perfectly tiles a 2D plane. This blew my mind. I wonder if it works with dimensions 5-10

[kpstl]

"Tetris induced fever dream" Matt Parker 2021

[KiloOscarZulu]

If 8D shapes cost 70 cents to make, Jeff Bezos has enough money to make every one of the 9D unfoldings! And have some money left over.

[stevegoodson]

This reminds me of an incident in my early education, when I was about 8 or so. Our teacher demonstrateed sticking together squares of paper and folding them into a cube, then just before a break asked us to think about how many different ways you could stick the squares together and still get a cube. I spent the break furiously scribbling away on graph paper, and after the break confidently strode back into the classroom and showed my answer - 20. The teacher took sadistic delight in pointing out to me that I'd come up with 9 pairs of reflected shapes, so 9 of my solutions didn't count. Spent the rest of the lesson passionately arguing that while rotations obviously weren't different shapes, reflections were. I think this was the point where my school realised I was going to be a problematic student to teach. Sadly I lost my intuition for maths over the years, would probably take me hours to come up with the wrong answer now.

[mrakoslav]

18:08 I went "I could find one", and I did! Thanks for the opportunity!!