Cardboard box tower

This is a cardboard box tower I made for my 2-year-old. Lots of secret doors, places to hide her Happyland people and little lights inside so it can be used as a nightlight in her bedroom at night. I also put lots of balcomies and places on it that she could put her people on as that seemed to be one of her favourite things to do as I was building it.

Detailed 2-point perspective cityscape

My latest and favourite!

(Click for a bigger image)

This one took me a long time! I've done other cityscapes before (see earlier posts), but I always had this one in mind. It took quite a lot of planning. I wanted a 2-point perspective composition where one point was on the horizon and the other in the nadir (directly below), but positioned in such as way that allowed me to have the head of the central building on the horizon and that also allowed a fair amount of space in the "upside down" sections at the bottom, because I think this gives more of a "vertigo" feeling to it when you first look at the picture. I also wanted the main tower off-centre so it encourages the viewer to "enjoy the view" as well as focusing on the main building.

So my first job was to work out how to build the construction lines, which I would then use as a guide when drawing the picture. This would have been easy if the nadir was in the middle of the picture, because perspective construction lines that radiate out from the nadir are straight if the nadir is in the centre of the circle. However because it's not in the centre in my case, the lines are curved... and I didn't know which way they curved! Here are some sketches, which include trying to work this out:

I couldn't find anything in my books or online that answered the question. I found a few things I didn't understand, and also some things that some people had put on the internet that turned out to be wrong even though it looked right, so that got me confused. I went as far as thinking that I was looking for a spectroscopic projection construction, but that turned out to be very wrong! After a lot of head-scratching and experimental sketches, I came to think of one of those traditional beach balls like this one:

Imagine cutting it in half across the middle (somehow without it deflating) and looking into one of the halves, so that you are looking at the central point (the white spot in the picture above where the coloured sections come together). Imagine this spot is the nadir and the lines separating the coloured sections are the perspective construction lines coming towards you. They will look straight if the white spot is in the centre. But as you rotate the bowl-shaped half-ball around slightly, the lines will curve. The way they curved was the answer to my question!

These perspective lines would later become the construction lines against which I would draw the vertical lines of my buildings. But I also realised that the lateral left/right perspective lines (which I would need to draw the horizontal lines on my buildings) can be drawn entirely within the circle, with their vanishing points on the limit of the circle, as in the very feint lines in the bottom left sketch in the photo below, (you have to click to expand the photo to see them) and not as I previously thought when I drew the bottom left sketch in the last sketchbook photo above.

As is often the case, I had over-complicated it and the answer was in fact very simple. So the next job was to design a tower that I was happy with. I drew literally hundreds of different variations until I did one I liked:

Finally, I had to set up my drawing in such a way that the vanishing points were going in the right place, which was a challenge, because the vanishing points are way off the page, on the edge of the circle (the horizon). So I taped some extra bits of paper around the paper I was going to use for the drawing and used a string tied to a pencil to draw the circle / horizon:

I then positioned the vanishing points around the edge of the circle and used this set-up to draw the construction lines, using big sweeps of my arm to get them as accurate as possible. This took a while because you obviously can't use a ruler, so I had to do a lot of rubbing out!!

After I had all the lines in, I got rid of the additional paper and started my drawing. The central tower was obviously the most important part, so I started there so that I could start again if I messed it up. Then I built the drawing up from there!

So there you have it! Comments welcome, but no requests for more of the same: this one took me just under a year from start to finish and I think my next drawing may only amount to a smiley face or something!

Media used: cartridge paper, pencil and Bic Atlantis biro only.

I think I know what Escher dreamed of

If you've looked at some of the other posts in this blog, you'll have noticed that I am a huge fan of Escher's work. Almost every single one of my drawings has been influenced by him. He was an incredibly inspirational graphic artist to me, but this isn't just because of his work in itself. Yes, his work is beautiful, it conforms with his idiosyncratic style and it approaches technical perfection (for someone who never used computers). But I'm mostly inspired by how he continued to innovate and come up with new ideas and themes to play with, and what he might have created if he wasn't stopped by his death in 1972.

So what were these ideas and themes? Well in my mind I classify them differently to how literature generally classifies them. In literature, you'll mainly read about four themes in Escher's work as follows: (1) traditional, normal face-value representations of people, animals and landscapes; (2) metamorphic transitional images and tessellations; (3) playing around with perspective; (4) approaches to infinity, (again with tessellations). I classify them differently because I'm more interested in his "break points". That is to say, points in his life at which he discovered a new idea and brought that idea to the world through his prints. I won't attempt to put them in any order, but here is my list. It doesn't include category (1) above, because this wasn't one of his "ideas". In my opinion, that phase was just a formative era on his journey towards the development of his own style.

(1) tessellations, i.e. shapes that fit together like his famous print, 'Sky and Water I' below. This section also encompasses both his metamorphosis period and his "approaches to infinity" period. The reason I've grouped them all together is because this whole phase was based on just one "device", namely working out how to fit shapes together and creatively repeating the patterns you can create this way.

Sky and Water I (1938)

(2) perspective under the rules of traditional Euclidean geometry, i.e. perspectives built upon straight, ruler-drawn construction lines disappearing to their vanishing points, like his 'Tower of Babel' below. This period of Escher's work was a fairly unpublicised one and as soon as he started experimenting with perspective in non-Euclidean geometry (my next bullet point), he looked back on this phase with regret. He said the following about Tower of Babel in a lecture when he was about 72: "I was 30 years old then and now find it to be a typical example of youthful clumsiness and lack of understanding of perspective." The number of works from this phase are limited because he soon discovered non-Euclidean geometry. (Euclid was a Greek mathematician: some say the Father of Geometry).

Tower of Babel (1928)

(3) perspective under the rules of "non-Euclidean geometry", as in a study he did for 'House of Stairs', below. Unlike Euclidean geometry, where perspectives are built upon straight lines disappearing to their vanishing points, non-Euclidean geometry observes that the human eye doesn't actually see straight lines as straight, it sees them as curved. The brain receives the information from the eye and interprets those curved lines as straight. We actually see things as in a fisheye perspective, but it took us about 2,000 years to figure out that the brain was doctoring this information while the poor old eye had it right all along! (The 2000 years is from 300 BC when Euclid laid out his observations, up until the early 19th century when the first of many other non-Euclidean geometries was discovered). This section also encompasses all of his fisheye perspectives. I myself am significantly influenced by this period of Escher's work.

Study for House of Stairs (1951)

(4) regular solids and symmetrical spatial structures, mainly comprised of polyhedra, Möbius strips, torus knots and so on. A good example of this is 'Stars' (1948) below. I'm using Stars as an example to illustrate this category, but I don't really like it because in my opinion, it serves as a somewhat crude and passionless 'display platter' of Escher's skills. However, despite this criticism, I am heavily influenced by this period of Escher's work as well, (which you can see from my blog, although I really like mandalas as well. Escher didn't really play with mandalas). I really love the way computers had no involvement in Escher's work here, and I'm confident that even if he were alive today, he would resist the channel of temptation that so many others have fallen into, namely using computers to construct, develop and inspire this kind of work.

Stars (1948)

(5) "impossible" repeating / cyclical perspectives. In other words, depicting things that are three dimensional but breaking the rules of spatial 3D geometry. This is arguably the theme for which Escher became most famous. Often, if people say they haven't heard of Escher, I describe 'Ascending and Descending' below, with its perpetually rising / ascending staircase, or 'Waterfall', also below, with its forever flowing cycle of water. They're famous because they're fun, interesting and pretty. Plus, they make good student posters! But I like them because they are good examples of how Escher kept on coming up with new ideas. Interestingly, when discussing Print Gallery (1956) with a group of revered mathematicians, Escher was commended on his innovative adaptation of an extremely complex mathematical manifold called a 'Riemann Surface'. Escher told them he had no idea what they were talking about. He said that if he had created a Riemann Surface then he was very pleased, but he assured them that he had done so by complete accident! This proves Escher's genius in the field of interpreting spatial geometry through art, not mathematics. (By the way, I understand that in its simplest linear form, a Riemann surface can be likened to the "Droste Effect". I blogged about the Droste effect in an earlier post, where you can also see Print Gallery and the Riemann surface he created).

Ascending and Descending (1960) and Waterfall (1961)

This final theme is especially important to me, and so significant, because it encompasses many fascinating ideas. It best describes my point about how Escher consistently and tirelessly devised, experimented, conceptualised and dreamed. In my understanding, this is what drove him. Not his physical tools or his media, nor his quest for mathematical accuracy, nor his earnings through the trade of his art. Rather, it was new ideas, new ways of distorting existing preconceptions and the biggest one: his quest to discover an entirely new type of spatial geometry. Just like Euclid and the geometers of the early 19th century, I believe Escher dreamed of discovering a whole new framework upon which a new view of the world could be built. He tragically died before he got there, but he said himself near the end of his life, "I could fill an entire second life with working on my prints". I think of this quote so frequently because I feel part of a collective consciousness of Escher fans around the world who, in their own small way, attempt to carry forward the flame that Escher held.

"Not one of us needs to doubt the existence of an unreal, subjective space. But personally I am not sure of the existence of a real, objective space. All our senses reveal only a subjective world to us; all we can do is think and possibly mean that therefore we can conclude the existence of an objective world" (Escher, 1953)

Mandala doodle 2

Cityscape

Stellated skeletal dodecahedra

For some reason I've grown very fond of stellated dodecahedra. Escher played around with this as one of many other regular polyhedra, but for some reason, this is my favourite to draw. I did this first picture on MS Paint to show how they are drawn. The rest were all freehand biro skeletal versions.

Mandala doodle 1

Samaritans volunteer doodle pad

These are photos I took of some of the A3 blotter pads in a Samaritan volunteer's phone booth. They get turned into massive collaborative doodles!

OK it's not technically (all) my own work but fascinating nevertheless, so I thought it deserved to go on the blog. I remember seeing a feature ages ago about these as well, in the glossy that comes with the Sunday Telegraph.

Click the individual pictures for bigger images, although the resolution isn't great unfortunately.

Doodles

Trefoil / Möbius Knots

A few trefoil knots I've drawn: all are just freehand and made with a black Bic biro (and pencil-line construction).

My most ambitious one yet, combining four parallel Möbius strips and a spiral tube running continuously round. This one was very challenging and took me a few days, because the fact that you can see through the structure meant that I had to show the back of it through the gaps in the front. The three cross-over sections were the trickiest, but I was pleased with the result overall!

A square-sectioned knot with hollow floating sections.

A square-sectioned knot with cube connectors. This one was quite easy because all elements are solid and opaque.

My first proper attempt at a trefoil knot, combining two interlocking parallel Möbius strips and circular ribs. The ribs aren't very evenly spaced, but the overall effect is pretty nice I think.