I have made a model of a cube out of well like paper tubes that are adequately stuck together
so I might be talking about something in a video like oh you know if you line it actually can I
line that up if I line it up it becomes a hexagon look at that hexagon ah the projection of a cube
corner to corner hexagon so let's say I want to build a model to demonstrate that and so I
put this together I'm like oh it's good just a bit difficult to transport if only I could reduce this
down have you know less cube while still having all of the cube and allow me to demonstrate what
I mean with a second cube so this in theory is exactly the same bar manufacturing imperfections
as the first cube we're going to keep that there as our whole cube and this one I'm going to cut in
half by just snipping four of these parallel edges like that okay so so now instead of traveling with
the whole cube I can just take this with me but Matt you think that doesn't look like the
original cube well I've got a mirror here so in the mirror first of all you can see Alex who is
filming me where are you Alex there they are hey all right so what I'm going to do is take half a
cube put it on the mirror suddenly if I hold it like that it's a whole cube again look at that
so there I've now got an entire cube made from half a cube and all I have to do is carry the
mirror with me but could we do better before we do do better a reminder if you want the physical
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up by like the 15th of December to get that thank you so much for all of your support right back to
past Matt and their planes of reflection I'm only using one of the planes of reflection of the cube
so if you cut all the way through here it's perfectly symmetric but there's not just that
direction we could do this direction as well but for that we'll need two pieces of mirror foreign
we have two pieces of acrylic very badly cut in half with a jigsaw and it means I can now
cut this in half again there and there now I just need to stick these together
we can now take a quarter of a cube pop it between these two perpendicular mirrors and there it is
we have an entire cube again check it out a whole cube but is that our only option well so far we're
only using the orthogonal planes of symmetry in a cube so like there's one that way there's one
that way there's one that way down each of the three um different axes however there's more
there's the diagonal ones so there's actually two diagonals every single direction there's a total
of nine planes of reflection for a cube so what if we wanted to combine some of those so I ran the
numbers on if we took the four diagonal planes of symmetry coming in for each of these four
edges at the top they all meet in the middle so I thought I could cut four triangles out to make
that little kind of inverted pyramid at the top and I cut them earlier today here are the bits I
just need to peel off the protective plastic stick them together see if it works foreign
I think that's it so I've stuck together the four triangles to form like if you go down that
way it's a pyramid it's actually that bit there from the top so I've actually I've designed it
a little bigger than this right so that means if I get a single one of the faces so just one
square I should in theory be able to pop that in there there it is and now I've got an entire Cube
okay I don't know how well that works from over there looks pretty amazing from where I'm sitting
that's a cube okay I will admit it's not perfect it's not bad
has a few flaws so first are all the edges are kind of double edges because each of the paper
tubes is itself reflected in the mirror so I could have you know meticulously cut them all in half
but technically they're you know infinitely thin lines if you ignore the double edges and
the alignment's a little wonky because uh look I just cut it I have a piece of acrylic with a
jigsaw which is not the most accurate way to do this you could get it laser cut you could get
it cut by someone who's not me but I wanted to work out if it could just be done at home just
give it a go buy some acrylic off the internet get a jigsaw see what happens and I think oh running
the numbers was fun so I started with a unit Edge on the cube and they did all the calculations how
far away is the center from the corner and all that jazz reasonably straightforward and you
end up with this very simple design I think it works pretty well however we're still only using
a single type of plane of reflection so you've got this is four of the kind of diagonal ones
that come across all merge together the one we had before this was two of the right angled ones put
together but now could we be even more efficient by combining the two and it turns out we can
yes
okay so I worked it out so this when I originally designed it I have made the mirror image of what
I designed that's fine still works it's all very symmetric so that should sit there so
that's touching the center of this Cube and then it's going up three different planes of
symmetry it's a bit hard to see but the one that's contacting an edge is this one over here that's
what's leaning against an edge so I need to do is get a tiny it's all I need imagine traveling with
that piece of cake you could take that and you're carrying on right so I'm going to put that lining
up with the same Edge in there like that and now when I look in there oh my goodness it's a cube
you're gonna have to trust me uh wait that's a bit I think
I've made it a bit smaller I'm gonna make it bigger so it sits higher up
there it is okay so you're gonna have to believe me totally looking at a cube right
now uh I thought that moves over the lineman oh my goodness that moves really easy okay I
gave up with the straw and I've got a little bit of Post-It note that I stuck to the mirror so it
doesn't move around you can see there's one Edge which you got there and then it continues up over
there so there's two two half Post-it notes make one Edge and then you can see a right angle in
the background there and then it goes across that way and then there's a right angle way back there
there's the next Edge over to there oops just have a shot is that there another right angle
comes over there right angle back to where we started so that's like the top face of the cube if
you as you look below that in the you can just see in the background there okay there you can kind
of see the other face going down from where it is there's that side on the side and it comes up over
there and you know it's the ones in the background are a little disjoint because technically you've
got like what's this this is 1 8 of a face oh my goodness so you've got a eight times six you've
got 48 of these that make up the whole Cube that's ridiculous and so yeah there it goes
that's around there around there oh my goodness but I think we can all agree uh on reflection
big success I've been casually interested in what I call kaleidoscopic geometry because it's a bit
like those old children kaleidoscope toys but I better say but much more precise give or take
I've been casually interested in these for a very long time actually five years ago oh my goodness
I was at the Museum of math Mo math in New York I was there for a meeting and at the back they
had some of these sorts of things set up and I put like a four second video up on my at the time very
new second Channel on YouTube showing how to flat pack a cube where I just comically pulled
the square out of the kaleidoscope and the whole Cube packed down into a square I was reminded how
amazing these mathematical kaleidoscopes are recently when I was at the bridges conference
in Finland that's a combined math art conference so much fun there were people there for from the
Museum of mathematics of Catalonia that's a maths Museum in Spain and they had taken
this in a whole new Direction so instead of something like this where you kind of get one
shape inefficiently or something like this where it's a bit more efficient they decided to see how
many different shapes could you get from the same mathematical Kaleidoscope with a priority placed
on the platonic solids those are the shapes where every single face is a regular 2D shape and every
single face is identical also all the vertices are identical and the Archimedean shapes that's
the same idea that you can mix and match different regular faces so instead of just having squares or
just having equilateral triangles you could have squares and equilateral triangles and they wanted
to see how many they could get into a single one and you can get more than one shape so when I was
messing around before when I put the straw in I put it in the wrong way and it looked like a
rhombic dodecahedron and I've messed around with it a bit more and where is it if I put it in this
way so I put it in a different direction on a different angle like this there it is I'm
now looking at a rhombic dodecahedron which oh here we go I happen to have totally by accident
I swear in the background you get one of these same symmetries as a cube isn't that amazing
because the symmetries are the same you can get it from the same Kaleidoscope in fact you can get
noctahedron if you're careful because it's the same symmetries as a cube and you don't have to
use every symmetry every time so you could have an abundance of symmetries and just use different
ones each time to get different shapes and that's what these folks did and they released a paper
I'll link to it below if you want to check it out it's called deltoid kaleidoscopes where they're
trying to work out what's the most like reflective shape bang for your back in a mathematical
kaleidoscope and they came across two very interesting ones there's the deltoidal I oh I get
this one wrong I Costa tetrahedral kaleidoscope and then there's the absolute winner where is it
the deltoid hexacontehedral Kaleidoscope that's one with 60 faces and when they built it it is
absolutely amazing here is the Kaleidoscope on display in the Art Gallery at Bridges 2022 and I
filmed myself looking inside of it and you can see that is an icosidodecahedra and regular pentagons
regular triangles check it out pretty amazing and it's all from that one kaleidoscope and you
can see next to it loads of other wooden pieces you can put them in and get different shapes
they've done an amazing job with this but only wanted one so I downloaded the paper looked at the
mass realized the geometry was quite difficult emailed my friend Sam heartburn and she said
she would render a lot for me and give me the Nets and here they are this is the one for the
hexachondrahedron and its four faces to make that Kaleidoscope uh two identical faces but you need
to do one reflection of each huh reflection it's a mirror and because she said do you also want the
other one I'm like yeah why not so she gave me the other one I cut these templates out drew them onto
more mirrored acrylic here are all the bits and so in theory I can stick these together and have two
more of these mathematical kaleidoscopes um now mine have not been as accurate as the other one
so I'm going to mix it up this time I'm going to peel off the protective film and then I'm going to
try taping because the tape is clear the inside of the mirrors which will make the alignment easier
but might mess up the reflections a bit but I think it's worth the risk let's give it a go okay
so what's great about these is as long as you print each one and its Mirror Image they only
go together one way it's the way where all the sides line up so that's the same length as that
so that has to go there it is that's the same length as that that's the same length as that
that's the same length as that okay so in theory the whole thing just kind of folds up now so I'm
going to stick them all like this fold them up stick them together to better explain what Sam
actually did I can recap what I did with a cube using the orthogonal plane of symmetry one of the
diagonal ones and then another diagonal one like a slopey down one on a different orientation and I
took the section that the three of them bound and that's what I made out of mirrors so I calculated
those three triangles built that gave me the rest of the cube Sam did the same thing with
this a tria contehedra which just means 60 faces I think of it as a dodecahedron where each face has
been split into five kites but there's other ways to look at it in fact as well as the decahedral
symmetries you get a bunch of other symmetries and so Sam took this kind of equator-ish line
of symmetry around there another one over here another one over there fourth one over there and
they bound this kind of wedge this pyramid going right into the center but instead of making this
one right to that final Apex Sam truncated it so we're just building the frost roomy bit of that or
rather that's what I'm trying to build oh good that one comes with a free fingerprint from me
all right oh well it's stuck there now isn't it I'm peeling that off again okay now in my version
I haven't got like a frame holding it in the rigid correct position so you just gotta flap it around
until it works I'm going to drop this Sharpie in I'm gonna see an amazing shape and you are
gonna believe me ah oh my goodness it works that's incredible oh if only you could see this somehow
I'm so happy right now I'm gonna make the other one actually I've realized because of the angles
that's to him very kindly designed if you put it down it naturally sits I mean you put it this way
flat packs that's handy there's a theme for the day if you put it this way and
you sit flat it automatically sits in the correct arrangement with all
the right angles you can then drop something in there it is
snub Cube rhombic dodecahedron although it's a bit of a shame like because the pens this is
boring colors if only I think this would be a lot easier to see you know it glow sticks
yes come on ah okay here we go ready
I'm so happy right now
and in the middle the wood the table of the wood makes the original Acosta tetrahedron shape
so what am I bringing to this field of making shapes in kaleidoscopes other than showing
that anyone can do it which If we're honest is frequently what I bring to the table but it's
important you can just grab the the plans I'll put a link to the files that Sam sent me print out all
four of each one put them on some acrylic cut them out stick them together you're in business anyone
can give it a go but on top of that I was thinking well now I've shown that you can tape these
together and you don't need external supports right it's literally just the mirrors and clear
tape I thought well it's a real shame you can only really see the shape in there if you look down
into the Kaleidoscope which I know sounds obvious but wouldn't it be great if the shape was a bit
more real and I thought hang on a second what if you could see through the side so I've also cut a
whole bunch of two-way mirror acrylic I'm going to peel off the protective film stick these together
put a glow stick in them and in theory I mean I've never tried this I don't think anyone has
in theory I'm imagining glow stick goes in turn the lights off Mass Rave right you should better
see the whole shape lit up because it's emitting like oh come on we're going to give it a go that's
how we're going to end the video little thing I've learned even though these are double-sided mirrors
and you think they can go either way only one face is technically mirrored and if you put that on the
inside the tape wraps it off so I mean because I thought it would mathematically be more precise
to have the reflective surface in the middle just practically it's not nearly as good so the second
one here I've put the reflective surface on the outside and the holes together a lot better but
anyway they're both built they both work let's turn the lights down awesome glow sticks in them
oh slow stick maths here it goes
by shape
ah it's there it's like a hologram floating in the air
it's not quite bright enough to go through the sides we're hanging closer oh it's coming up
like a gateway to another world oops
and put that one on top of it the different sizes put that one at the bottom
yep
that one's the snub cue foreign