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ORGANIC TOWNS FROM SQUARE TILES - a talk by OSKAR STÅLBERG at INDIECADE EUROPE 2019 — Transcript

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4:25let's start okay it is me again so our
4:33next speaker is Oscar who will show you
4:36how the wavefunction collapses I told
4:39him I'm gonna make a joke so I had to
4:40please welcome Oscar oh now it's working
4:51yeah hello everyone yeah I'm Oscar
4:53Stalberg I'll be talking to you about my
4:57new project that looks like this that I
4:59am currently working on the talk will be
5:02quite technical probably more technical
5:03than a lot of talk so far but I'll start
5:06by just introducing myself
5:07my name is Oscar Stalberg I've worked in
5:12video games for like 7 years now I think
5:15I started out at Ubisoft massive so my
5:17first part is I worked on was the
5:19division where I made this cool
5:21holographic map and sort of maps and
5:23urban environments has been a big part
5:25of my work ever since in the middle here
5:29you see two like web demos I did a long
5:31time ago those are accessible like
5:33online you can play them in browser and
5:35build your own little house or build
5:36your own mobile planet so I'm very
5:39interested in procedural stuff in
5:41general and that's the talk is gonna be
5:42a lot about procedural stuff and the
5:45last gift you see er here is from the
5:48game bad north which is what I like the
5:50past two years I was working on the game
5:52bad north so it's a small procedurally
5:57generated micro strategy game where you
6:00defend Islands against Vikings that also
6:04started as a procedural project like all
6:07my projects do and then I was kind of at
6:09a good place in my career where I
6:10figured well might as well put a game on
6:13top of this and like do my first indie
6:15project yeah but that's all out now it
6:17actually shipped on mobile two days ago
6:19so now we've shipped on all the
6:21platforms that we were gonna ship on
6:24yeah and if you want to see cool gifts
6:27like this you could follow me on Twitter
6:28that's my Twitter handle right there but
6:31today I'm going to talk about my new
6:33project it doesn't have a name yet the
6:36working title is placed
6:39because you're building nice little
6:40places kind of but this is what it looks
6:43like
6:43and it's kind of the it's kind of a
6:46combination of a lot of themes I have
6:49been working on lately so it uses a lot
6:53of the same core ideas from bad north
6:56but it also combines it with some with
6:58things from brick block that thing where
7:00you build a house and a bunch of other
7:03stuff and then it introduces the new way
7:05of doing a like a non-square grid you
7:07can see so we can create kind of more
7:08organic looking tiles which is something
7:11I've been wanting to trying out try out
7:12for a very long time yeah so i'ma talk
7:17about just like the different techniques
7:19that go into making this thing so so
7:24first of all what is it its a mix
7:27between the algorithms marching cubes
7:29which is quite a simple algorithm but
7:31like that's a good first procedural
7:34algorithm to implement and the algorithm
7:36wavefunction collapse which is a bit
7:37weirdly named algorithm but it's an
7:39algorithm for like placing tiles or
7:41inside a space as its a mix between
7:44those and then it's also running on an
7:48irregular quadrilateral grid so
7:51quadrilateral means that all the like
7:54you see how the little black grid here
7:56it's all made of squares or not squares
7:59it's made of quads right so they all
8:01have four corners but it's irregular
8:04right here you have three quads meeting
8:08at a corner and you can build there if
8:10you want to or here you have five of
8:12them being a corner and you can probably
8:14yeah here's there's six of them meeting
8:16at all that's too far away I can build
8:18like that there we go so I'm gonna start
8:21to talk just about how I am approaching
8:24the generation of a grid like this it
8:28took me some time to like I knew I'd
8:30been wanting to work on a grid similar
8:32to that for some time but it took me a
8:34while to figure out the best approach
8:35for it so imma just show you a get like
8:38some of you might have seen this gif on
8:40Twitter before so that was the only
8:42slide now it's just gonna be in unity
8:44and a couple of pictures
8:47so here you can see how the grid is made
8:51so I start by creating a bunch of
8:53triangles within a hexagon and then I
8:56randomly join some of the triangles into
8:59quads so that's the only random element
9:02in creating the grid because after that
9:05like the grid like if you and I just see
9:07that with the position of that
9:08particular grid patch then you know it's
9:10deterministic yeah so then it randomly
9:14joined some of the triangles into quads
9:15and then it subdivides all the quads and
9:19all the triangles because obviously you
9:21can subdivide subdivide a quad into four
9:25new quads but you can also subdivide a
9:26triangle into three quads so it's like
9:30if you run the smoothing algorithm
9:31function in any 3d software that's
9:34basically what it does and then I run a
9:36relax algorithm on that to make it all
9:39kind of smooth and nice and not like
9:40jagged and sharp yeah so I do that and
9:45the point of that kind of is the point
9:48of using a quadrilateral grid like that
9:50well the point of using it rather than a
9:53just a square grid is of course because
9:55like town escapes look more fun when
9:57they have fun and interesting organic
9:59shapes I mean Pirates of all towns kind
10:01of look like this a lot but it's also
10:04because if we have if it's all made of
10:05quads then you can use square tiles and
10:08place them there so that means it's
10:10quite easy to input like that's a good
10:12functional you can like you can
10:13tessellate square so you can place them
10:15next to each other they have edges that
10:17bump up against each other so it's like
10:19that's how we normally use tiles and
10:20games so it's just a more fun way of
10:22using pretty regular game tiles actually
10:26the first actually did some work for
10:29those who game released of like a few
10:32months ago called night call that takes
10:34place in Paris you're a taxi driver
10:36driving around and they just needed a
10:37view of Paris like you're driving around
10:39on the actual map of Paris and they just
10:41needed a view of Paris
10:43this sort of swish by in the rear view
10:44mirror as you're driving around so bait
10:46so they contacted me to help them out
10:48with that and that was my first attempt
10:50to try out making a quadrilateral grid
10:52so this is what I did for that one so
10:54you can see how like the grid it or the
10:56here it's actually generating it within
11:00city blocks that are like the actual
11:02blocks from actual data of Paris but
11:07that was pretty complicated to write
11:09because when you and you can see how it
11:11gets kind of wonky around a lot of
11:13especially like sharper places but for
11:15my new project and I wanted to figure
11:17out is and there's a lot of like I have
11:18to do a lot of intersection tests
11:20because you're kind of tiling it up
11:21you're like you have is an irregular
11:23polygon an arbitrary regular polygon and
11:25then you're tiling it up with quads
11:27along the edges and then you sort of
11:28keep filling it in until you filled it
11:29in all the way but if you make a mistake
11:31and the quads start sort of overlap each
11:34other then it will all turn inside out
11:36and then it will start building towards
11:38infinity and you obviously don't want
11:39that so that was quite tricky and you
11:42have to do a lot of just line line
11:43intersections and quad quad
11:44intersections check like can I play
11:46something here so in this new project
11:48I'm making I want it to be very
11:50performant and be able to run like in
11:51the browser or in mobile and stuff like
11:53that so then I wanted a solution that
11:55didn't contain where I didn't have to do
11:57any intersection tests and stuff like
11:59that so I'm very happy that I bet I
12:02figured out this model yeah and we can
12:06actually see let me show you it
12:08generating on the fly so now the game is
12:09running so we'll turn off the game and
12:11then we will turn that like I can tell
12:15the game how fast its update loop is
12:17supposed to run so now it's quite slow
12:19it's just going to be able to do eight
12:20steps each frame normally I run like a
12:26couple of like eight milliseconds each
12:27frame so we'll see the yeah here you can
12:30see how like it's building everything
12:33and the sort of tiles are slowly falling
12:37into place there's still some
12:38awkwardness but that will sort of soon
12:46and all these like red things are
12:49marking out like where it's supposed to
12:51generate new because now it's just
12:52generated a little grid patch in the
12:53middle here it's actually up to speed
13:01here a little bit yep so now you can see
13:03it like now it's generating these little
13:06grid patches and like a challenge here
13:08was also you can't like a relaxation
13:12algorithm you can only run a relaxation
13:13algorithm on a finite set of points but
13:16I wanted this to be a erratically
13:17infinite grid where you could just keep
13:19building forever I mean obviously you
13:21will run out of memory but you can like
13:22build a bit here and then remove those
13:24things build them here remove those
13:25things built-in mirrors you can kind of
13:26walk by building in different directions
13:28like if I place if a place a tile like
13:32here it will start to build like build
13:35grid in this direction too so what I
13:37have to do is I have to generate a bunch
13:39of these little patches so here you can
13:42see the clear like the the patches made
13:44of triangles that are subdivided that
13:46turn into quads but I have to generate
13:48clusters of those patches that I then
13:50can sew in clusters of three as you can
13:53see here with this thing that kind of
13:54bends into place I relax them together
13:57and then with the final result I can
13:59interpolate between a bunch of different
14:01of those relaxed clusters and then you
14:03get like a seamless tiling thing made
14:06out of quads so you can see how like
14:08here's obviously the borders between the
14:10different patches and you can see how it
14:11all lines up perfectly in the end yeah
14:17so now it's generated like everything it
14:19has to generate for my current size of
14:21the world but if I keep building this
14:22way yeah you see how now like it wants
14:24to generate more in this direction and a
14:26bit more in this because there's like oh
14:27yeah
14:29so and a very interesting thing in
14:31generating the quad grid this way in
14:34starting with generating triangles is I
14:37got an interesting opportunity use
14:38something I've been wanting to use for a
14:40long time which is there is a way of
14:43thinking about hexagons like has any of
14:46you ever built a hexagon based games
14:48yeah few knots yeah okay a few people
14:51there is a very interesting way of
14:53thinking about hexagons where instead of
14:56where you can think about them as as
14:58cubes in a sort of diagonal plane
15:01stacked like this because as you can see
15:04here these are cubes but they're also
15:05hexagons and then you can use integers
15:08instead of floats for their positions
15:12you don't get precision errors and you
15:13can do like a lot of the math becomes
15:14much more straightforward than if you're
15:16using positions and angles and stuff
15:18like that then you get a lot of floats
15:20and then if you build a very large grid
15:21you start getting flowing points issues
15:23and stuff like that so it's a very nice
15:25way to approach hexagons yeah and it
15:31also makes sure that the grid kind of
15:33has this nice irregular shape right so
15:36none of these not like there isn't an
15:38underlying square grid that creates this
15:40this quadrant but but it's all made of
15:42hexagons yeah another fun thing with the
15:46hexagons is oh yeah and if you want to
15:49look further into there's an amazing
15:50article that I've used a bunch of times
15:52written by like the addresses here by
15:54Amit Patel he does a bunch of I mean I'm
15:56sure you've come across some of his work
15:57before he does a lot of really
15:59interesting really well explained sort
16:03of articles and blog posts about some
16:04fundamental like path finding algorithms
16:07ways to structure your grids graphs
16:11stuff like that and like procedural
16:13generation stuff in general so I got a
16:15lot of my hexagon math from him yeah and
16:23an interesting way to illustrate this as
16:25well is that since all the verts here
16:28have an origin position on this like as
16:32integers I can actually use I'll show
16:36you the bounding box I'm using to decide
16:37like how what's the center of this world
16:39is and how far out I'm going to generate
16:40so this looks like that and it looks
16:42like a kind of a hexagon but if we build
16:45it a lot in one direction you can see
16:47that it acts it is actually a box but
16:50that's projected down into the flat
16:51plane into the hexagon space so it
16:54becomes a bounding box but in in hexagon
16:58space which means that you get less of
17:00the obvious bounding box center that you
17:03would get if it was just a grid aligned
17:07bounding box the normal way you would do
17:09it so that's a nice little illustration
17:12of what's
17:14yeah so that's kind of it about know
17:17there's a few more things about the grid
17:20so yeah obviously when I run the relax
17:22algorithm to make the grid look as as
17:25sort of nice and relaxed and smooth as
17:27possible I found an interesting way of
17:28doing that and that could be a general
17:30way if you have like an irregular quad
17:31and you want to turn that quad into more
17:33of a square but you don't want to align
17:35it to the the axis of the of the world
17:40you're working and this is very
17:41convenient and fast an easy way to do
17:42that where so you start by like
17:44calculating the center and you get the
17:46vectors like from each corner to the
17:49center right and then if you just rotate
17:52them all like rotate one of them nothing
17:55and 190 degrees one one eighty degrees
17:57and one 270 degrees and the way you can
17:59rotate a 2d vector is if you just flip
18:01the you flip the coordinates and it
18:05invert one of them that's a 90 degree
18:07rotation of a 2d vector so that's a
18:08super cheap way of doing it so you
18:10rotate them and then you average them
18:11and then you rotate them back out and
18:13then you just move towards that right so
18:15then you get a perfect square in the end
18:16of course in my relaxation I can't make
18:19a perfect square out of them because
18:20like I could make a perfect square out
18:22of one grid but since they're all
18:23budding up or one quad but since they're
18:25all budding up against each other this
18:27I'm sort of accumulating these little
18:29forces like they sees a little red green
18:31blue yellow force on these corners I'm
18:34accumulating them from all the like for
18:36each verse for all the adjacent quads
18:38and then I sum them up and then I do
18:39that over multiple iterations and that's
18:42kind of the slowest parts of the with
18:45one of slowest algorithms that I have
18:47running the relaxation because you have
18:50to do it quite a lot of times like when
18:51you saw the grid relax in the beginning
18:54you can see how like it actually takes a
18:56while to relax all of them and it keeps
18:57like it's still having effective and 256
19:00steps out which is how fast I'm running
19:02it yeah yeah so that's the the grid I'm
19:07working on another interesting thing I
19:09do with the grid is I I want to create
19:12like sort of different areas that are
19:14dif work aortic and some feel more
19:18orderly so you can see how some of these
19:22some of these
19:25hexxus are like this here you can
19:28clearly see the remnants of a triangle
19:30right and here's clearly the remnant of
19:32a triangle
19:33whereas this piece here is just made out
19:38of six very big triangles so this piece
19:41becomes much less chaotic than this one
19:43here that's made out of more anyway and
19:46this one here is made of a ton of small
19:48little triangles like you can see how
19:49the triangles is kind of made up of
19:50learn or Mike here's the mid-level
19:54here's the very very big triangles and
20:01yeah this is the more chaotic one so
20:04these end up like if you build around
20:05those areas you get a much more chaotic
20:08looking town with like weird Wonka
20:10shaped houses like you see like that's a
20:12weird
20:14that's a weird house shape obviously
20:16whereas if you build in the more axis
20:21aligned areas you can build nice big
20:23square ish function so that's kind of an
20:25option I want to I want to give the user
20:26in the end to kind of choose what kind
20:28of style they want to build their
20:29environment in but I don't want them to
20:31be able to build a completely square
20:33thing too big like they're always
20:34supposed to butt up against the sort of
20:36organic nature of a city and that's kind
20:41of one of the fundamental sort of
20:43aesthetical things of this project is I
20:46really like the in urban environments I
20:48really like the combination of the
20:51artificial and the plant and the
20:53symmetrical which is like whenever you
20:55build a house usually well not anymore
20:56but it used to be did you build it kind
20:58of symmetrical with a clear sort of idea
21:00and repetition and patterns but then you
21:02know it always butts up against another
21:04house was sort of a weird angles and you
21:05have to make it a bit of a weird angle
21:06and then after a while when you're
21:08renovated you've changed like half of
21:10the house likely or so in the end you
21:11have this sort of organic asymmetry
21:13growing so new symmetrical things keep
21:15popping up but then they kind of become
21:17more organic and symmetrical less the
21:18environment grows around them and that's
21:20kind of a thing that I wanna an
21:22aesthetic that I want to that I want to
21:24work with in this game I'm hoping to add
21:26later like right now I just have these
21:27little grassy patches but I'm hoping to
21:29add later stuff like plants that
21:32actually grow slowly over time like
21:34vines growing on the walls would
21:35obviously be ideal but even just a
21:38little tree
21:38or whatever and stuff like that it kind
21:40of grows slowly over time so house
21:43obviously this is a very tile based
21:45thing so if you just look at the how it
21:49all comes together and turn off the grid
21:51of this grid - so these are here you can
21:54see how it's all divided into little
21:56tiles and these I for this project I
22:00have to build a lot of tiles to make the
22:02make the whole thing work so this is
22:05actually what it looks like in Maya when
22:06I'm building them so that's quite a few
22:08tiles I'm up to about 200 tiles already
22:11but I reckon I'll have to build like 3
22:12or 400 to satisfy all the possibilities
22:15of what you whatever you can build and
22:17so these tiles look very similar to
22:20something that you would use for a
22:21normal marching squares algorithm and
22:23one core one core idea is that the tiles
22:27sit on kind of the corners right so if
22:31we select a small little house piece
22:33here so this is just a single block of
22:35house right but it's made up out of like
22:38four corners on top and then four
22:41corners on the bottom like this so each
22:44of these little modules have like this
22:49one obviously only has one corner that
22:50sort of inside the house block whereas
22:53it has like the rest of the corners are
22:54outside the house book so we can look at
22:57what that looks like so here is all
23:01these tiles imported into Maya but they
23:05don't have a texture here this this mesh
23:07here is actually just for debug purposes
23:08for me to be able to see what I'm
23:09working on
23:10and here incidentally is actually the
23:12different tiles I'm building being
23:14recombined into bigger tiles because I
23:16don't wanna have to build all of these
23:18weird combinations myself so whenever
23:21you have tiles that could fit it like
23:22you could have a corner like this
23:23against the corner like that and they're
23:25not intersecting I might as well just
23:26generate that out of the meshes I
23:28already have instead of building it
23:30manually because there's a lot of
23:31combinations to build obviously this
23:33like goes into forever here and here
23:37also you can see how let's pick the
23:40thing we were looking at before here's
23:42that little thing so if we select this
23:44one here I should be able to click on it
23:45and yeah oh it's a bit slow when I try
23:50and display all those so you can see how
23:52this how it
23:52little square on the or a little cube on
23:55the corner here that's because that's
23:56the only cube that's inside whereas if I
23:59click this piece of wall here it has
24:01four cubes because it has four corners
24:03that are inside and it has four cubes
24:05that are outside so that means of course
24:08that you can like taking those corners
24:12in sequence you can combine them into
24:13right if it was just inside and outside
24:15it's not that simple but if it was just
24:17insides and outsides you could you could
24:19put it into just a binary number right
24:21so this one here could be this one here
24:25could be one zero zero zero zero zero
24:28zero zero so that's like the byte that's
24:30one or if the corner was placed here
24:32instead of here then it's going to be 0
24:341 0 0 0 0 0 which is 2 so that's a great
24:37way you can just index them like that
24:38and then when you ask for them again and
24:40the algorithm is very easy to just ask
24:42access the right the right one straight
24:43away now in my case there is one pretty
24:47significant piece of complexity above
24:51that which is that I want to be able to
24:53use multiple colors so if I build let's
24:57run the game again if I build with more
25:02than one life I build with like color
25:04colors of blocks of different colors
25:06next to each other I want to like make a
25:08little comment in the mesh of what
25:09happens when those two blocks meet right
25:11so here you can see I'll let's make it
25:13run faster there we go
25:16so here you can see how this green meets
25:17the yellow so then there's like a little
25:19trimming like this and it has like a
25:21slightly taller window and when these
25:23ones meet each other like that there's a
25:25little bit of a drainpipe yeah basically
25:28I want to be able to and here these meet
25:29and they have like slightly different
25:30heights so I want to be able to comment
25:32what happens when different colors meet
25:34but that also means that I have to like
25:36it's not enough to have a court like a
25:39house corner like this I need to have a
25:41corner where like the top or house side
25:44like that I need to have one where the
25:45top part is one material color and the
25:48bottom one is another material they
25:51don't this doesn't all only work for a
25:53green and white though like whichever
25:55color you choose you can pick this one
25:56and they recolor them offers so I have
25:59to do that one but then I also have to
26:01do like ok what happens if we split the
26:02colors this way around and what happens
26:04if we split the colors
26:06like here's one that splits the name for
26:07different ways so then I have to like
26:10assign different colors and build all
26:11these kinds of different combinations so
26:12that's a lot of work and that's why I
26:14think I'll need another if I didn't have
26:16the different colors I think I only
26:17would theoretically need 667 pieces to
26:20make this whole thing work but with the
26:21different colors the permutation count
26:23kind of explodes so now I think it will
26:25be like in the three four hundreds or
26:27something like that yeah so the marching
26:32squares part the marching cubes part of
26:35the algorithm that's the simple part
26:36that's the part that can figure out that
26:39you know but this is a corner and can
26:43place the corner on the corner of a
26:44house but that's not all it takes
26:46because obviously I have a bunch of
26:47different types of corners right so I
26:49have this top corner here it's like a
26:51simple type I have now became a
26:53different type but then of course I have
26:54like this little corner here where it's
26:56like a terrace or this little thing here
26:59where it's like a bigger roof or than
27:02those all kinds of things like that so
27:03that's where the wavefunction collapse
27:05algorithm comes into place so let's see
27:07if we can actually see this unfold in
27:09real time so if I turn on the gizmos I
27:11select the that's the graph I need to
27:16select and then turn on off the graph
27:17gives me turn on the cube there we go
27:20so nope that's the wrong one yeah okay I
27:22think this should work yes
27:24so here it became very very slow and now
27:26it's the drawing of everything that's
27:27very small so what you see here the
27:29white wireframes you see is the entire
27:33possibility space right so for each of
27:35these little slots it there is there a
27:42bunch of different modules that are
27:43possible but it's usually not that many
27:45like on a corner slots maybe there's
27:47like I don't know 20 different ones
27:49because it's like a few different styles
27:51but then they can be rotated and mirror
27:53different ways so it starts with all of
27:56those being so let's just maybe look at
28:03its bit slow if you have everything
28:04selected so let's maybe look at them
28:07here I think it's still yeah so here if
28:09you look at this corner you can see kind
28:11of the shadows of let's turn on the word
28:12from the shadows of the different
28:15possibilities you see like it could have
28:16a chimney here it still thinks it could
28:19have a like a pointy side and achieve me
28:22here it actually can't in the end
28:23because that would there would have to
28:25be a similar chimney on this side and
28:27there's not but it doesn't know that yet
28:29because the algorithm hasn't gotten that
28:30far yet right so right now this thing's
28:33that there are eleven possibilities for
28:36this cube and those are the ones we see
28:38here and this one things that are that
28:40there are ten possibilities for this one
28:42this only thinks there are eight like
28:45you can see the shadow of the the kind
28:48of trimming we were looking at before
28:49here but as to then ask the wavefunction
28:52collapse algorithm runs fewer and fewer
28:55things are possible and in the end it
28:56all collapses so if you're just running
28:59again we can if we select everything we
29:01see all the gizmos for it yeah and the
29:07so obviously the wavefunction collapse
29:09is an algorithm that can be quite slow
29:11I've managed to build my thing in a way
29:12so that I can run it like if we run it
29:16very slow and I can keep clicking and
29:19even though it hasn't finished figuring
29:21out what things are supposed to look
29:22like I can still keep building and then
29:25it will just like if we just make it
29:26faster again okay now I selected it so
29:29now it's very slow then it will kind of
29:32resolve it all anyway after the fact so
29:37there's like two kinds of propagation
29:39happening first there's a propagation
29:40where like if you build something it
29:43needs to propagate out and tell all its
29:44neighbors like hey something changed so
29:46everything is possible again and that
29:49one tells its neighbor hey I changed and
29:50that means I could have a different side
29:52towards you so that means everything's
29:53possible for you again and that kind of
29:55propagates out obviously if you have
29:56disconnected houses like this it won't
29:59propagate to them so if I select all of
30:02these and build here it might even be so
30:06fast that yeah cuz if I build here you
30:08see how this entire chunk it becomes
30:10like everything becomes possible again
30:11and then the possibility space collapses
30:13and yeah where I said but if I build
30:15over here these aren't connected to this
30:17but if I connect them like this then
30:19yeah that all happens so that I have
30:23done a longer totten like hour-long talk
30:25on just the topic of wavefunction
30:27collapse for anyone who wants to dive
30:28deeper into that so that's the name
30:30again of the algorithm that kind of
30:31figures out which pieces fit next to
30:33each other and select
30:33them but the core idea is like you make
30:36you build a large possibility space
30:38where everything is possible in my case
30:40here not everything is possible because
30:42just the right like corner pieces or
30:44side pieces that satisfy the constraints
30:45around they are possible and then you
30:49start removing you start as soon as you
30:51place this one piece that big means that
30:52a bunch of the pieces next to them are
30:53not possible anymore so then you start
30:55removing those and then that means that
30:56a bunch of other ones are not possible
30:58anymore and that propagates and the
30:59whole thing collapses and in the end no
31:01it's just one possibility and that's
31:03what you end up showing the player yeah
31:07that went by way quicker than I thought
31:10because I have a bunch of other things
31:11but maybe we'll cut for questions to
31:14wrap it up maybe yeah ok but all I'll
31:17have this project running on my computer
31:20I will be sitting around somewhere so if
31:21anyone want has further questions or
31:23want to take a look at anything or stuff
31:25like that
31:25then you are most welcome to approach me
31:27thank you
31:29[Applause]
31:53you

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