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Dali and the Fourth Dimension: "Impossible" Objects, Part II

Posted March 08, 2013 10:15 AM by Hannes

In the first Impossible Objects installment, we investigated the cyclical cross-pollination of mathematics and art as Penrose and Escher inspired and re-inspired each other. In this post we'll delve into another, more eccentric trio: Salvador Dali, Thomas Banchoff, and the fourth dimension.

Conceptualizing dimensions higher than the third is not a new concept. Mathematicians began kicking ideas around in the early 19th century, with the relatively "big names" of MĂ¶bius, Riemann, and Hinton taking stabs at fourth-dimensional geometric figures and arithmetic. The 1884 novella Flatland (a fantastic little read, in my opinion) explores and compares higher dimensions from the perspective of its narrator, a two-dimensional square. Flatland is a prime example of using dimensional analogy, or the study of how a dimension n relates to its neighboring dimensions, (n-1) and (n+1). (So, in the case of n=3 [the third dimension], the comparison of the third dimension to the second and fourth.)

To the outside observer, Salvador Dali - the guy who did the limp watches, sliced the eyeball, and was never spotted without his waxed mustache, billowing cape, and pet ocelot - was always in search of novel methods of stretching the physical world. In 1954, Dali painted Corpus Hypercubus, a quintessentially Dali-esque amalgam of religion, mathematics, and physics. While this painting may appear to be a representation of Jesus hanging from a geometrically-altered cross, it's actually an example of dimensional analogy: the "cross" is an "unfolded" tesseract, a fourth dimensional analog of a cube. (Schlegel's model of a tesseract is shown below.) Just as it's possible to unfold a three-dimensional cube into six squares, it's also feasible to morph a tesseract (using the n-1 dimensional analogy; fourth to third) into eight cubes.

Dali, as well as Captain Marvel comics, inspired a young student named Thomas Banchoff - now a professor of mathematics at Brown University - to devote his life to the study of the fourth dimension. Using Corpus Hypercubus as a model, Banchoff constructed a paper model of an unfolded tesseract and was photographed with both the model and Dali's painting by the Washington Post in 1975. A few weeks later, Dali - who resided in New York at the time - dropped Banchoff a note requesting to meet with him. Nervously assuming the request was either a hoax or a follow up on the illegal use of Dali's work in the Post, Banchoff was surprised to discover that Dali was interested in meeting with him to pick his brain about representing higher dimensions. The two met several times, became fast friends, and corresponded often for the rest of Dali's life.

Dali was known for his respect for and friendships with mathematicians and scientists, going so far as to say that, "Scientists give me everything, even the immortality of the soul." We've all had that one weird, out-there friend, but Dali? I'd say he takes the cake.

(Image credits: Wikipedia | The peacock's tail)

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#1

Re: Dali and the Fourth Dimension: "Impossible" Objects, Part II

03/08/2013 11:12 AM

Robert Heinlein's short story "And He Built a Crooked House" is sort of a guided tour of a tesseract.

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#2

Re: Dali and the Fourth Dimension: "Impossible" Objects, Part II

03/08/2013 1:24 PM

There is a fantastic museum dedicated just to Salvador Dali right on the waterfront in downtown St. Petersburg, Florida. It's an interesting building at a very scenic location. Worth a visit if you're ever in the Tampa - Clearwater - St. Petersburg area.

The Dali

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#10

Re: Dali and the Fourth Dimension: "Impossible" Objects, Part II

03/09/2013 8:29 PM

Went there recently. When I got to the parking lot and stepped out of the car my first thought was the building was designed by cubists!

My favorite work of his is "The Discovery of America by Christopher Columbus".

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#11

Re: Dali and the Fourth Dimension: "Impossible" Objects, Part II

03/09/2013 9:36 PM

I went there years ago, long before the new building was built. I always liked the variations on the 'Persistence of Memory' with the melted clocks. I also liked the digital Abraham Lincoln one, though I can't remember if I saw it there or in a book about his work.

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#12

Re: Dali and the Fourth Dimension: "Impossible" Objects, Part II

03/09/2013 10:52 PM

The thing you can't pull out of the pictures you see in the text books and magazines is the extraordinary minute detail in his paintings.

Even his large works have very fine resolution in the details.

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#13

Re: Dali and the Fourth Dimension: "Impossible" Objects, Part II

03/09/2013 11:00 PM

I had the fortune to see "The Artist's Eye" at the Henry Gallery in Seattle. The detail was indeed extraordinary. I don't recall this painting being reproduced in a few books I have about Dali's work, so it was a most delightful surprise.

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#3

Re: Dali and the Fourth Dimension: "Impossible" Objects, Part II

03/08/2013 11:08 PM

I recall having pondered on these sort of 4th dimensional approximations years ago in high school. (I had lots of free time in study hall.)

Just adding more 3D boxes to the original box at odd angles is a rather poor representation of the 4th dimension in my opinion.

To me adding the 4th dimension of time does not change the shape of an item it only adds that concept of when to it. The X,Y,Z reference points, and distances relative to each other if any, stay the same but given the T dimension they are only going to be potentially offset relative to all common reference points by the distance changes, if any, that occurred over the additional dimensional unit T.

X,Y,Z +- XT,YT,ZT sort of.

Well it make sense to me better that a bunch of additional oddly angled and positioned cubes does any way.

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#4

Re: Dali and the Fourth Dimension: "Impossible" Objects, Part II

03/09/2013 1:37 AM

I'm with you. I would say viewing a slow-motion video of a cube falling or maybe a water droplet would suit me better. But here are a couple of videos I pulled up to check out all the same. They might give you a different perspective on the subject. Then again, they might not

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#5

Re: Dali and the Fourth Dimension: "Impossible" Objects, Part II

03/09/2013 8:07 AM

The tessaract is a 2D representation of a spatial 4D 'cube'. It's WXYZ, not XYZT.

In a 3D cube, the faces are squares that are connected along one edge, with each face being 'rotated' 90 degrees into a 3rd dimension that does not exist in the 2D world of the square. A sixth square parallel to the first one, but displaced through this 3rd dimension completes the cube. Some distortion occurs when drawing a 3D cube onto a 2D plane.

So for the tessaract, each cube is joined along one face to another cube, but each cube is 'rotated' through 90 degrees into a 4th spatial dimension that does not exist in the 3D world of the cube. Passing from one cube to the next requires a 90 degree rotation through that 4th dimension. An 8th cube, 'parallel' to the first one, but displace through this 4th dimension completes the tessaract. A lot of distortion occurs when drawing a 4D tessaract onto a 2D plane, including having the 8th cube appear much larger than the original cube. To a 4D being, all the cubes would look the same size.

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#8

Re: Dali and the Fourth Dimension: "Impossible" Objects, Part II

03/09/2013 2:45 PM

I guess it all comes down to what the fourth dimension is going be defined as. To me its based on time which is something we can perceive yet is not one the three primary dimensions we interpret as our reality but has an observable effect and influence on everything associated with the other three as we perceive them.

To me adding extra theoretical to us physical dimensions fits into the 5th 6th and so on dimensional aspects since they would be extend parts of our primary three physical ones but not necessarily moved through the fourth that we can perceive being ethereal the point in time dimension.

-------------

My wife just asked me what I am doing and I told her I am studying up on the fourth dimension of time.

I said, because I want to finish my time machine.

She then asked, well do you have it working yet?

I said, Well you're still here aren't you? So the answer is obviously no!

------------

I find that joke relatively funny!

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#14

Re: Dali and the Fourth Dimension: "Impossible" Objects, Part II

03/12/2013 6:04 AM

Hi Usbport. Good presentation about the "hyper-cube" issue (you call it "tessaract", I prefer to call it "hyper-cube", as it is a geometrical object located in the 4D hyper-space). However, I'd like to add a comment-correction.

What you have shown as a hyper-cube presentation on a 3D space -i.e. the "cube inside the cube"- is not the geometrical presentation -i.e. the geometrical projection- of a hyper-cube on a 3D space, but actually it is the optical projection of a hyper-cube on a 3D space. I made some drawings to show you that.

Consider a cube located above a plane as shown in the Figure1 (i.e. a cube's face is parallel to the plane). Consider this cube as a "construction made of wires". Above this cube there is a light source. As the light falls on this cube, you'll see the cube's shadow on the plane. This shadow is shown in Figure2 -i.e. a "square inside a square"- and this is, actually, the optical projection of the cube on the plane. In the same way, what you have shown as a "cube inside a cube", is actually the optical projection of the Hyper-cube on a 3D space.

So, in order to be precise, as the actual geometrical projection of a cube on a plane (i.e. an image on a plane) is like that:

then the respective, actual geometrical projection of a hyper-cube on a 3D space (i.e. imagine it as a construction on a 3D space) is like in Figure3:

Figure4 shows the way to design such a shape. There are two cubes located on the 3D space, shifted to each other, with their respective edges connected. The Hyper-cube is combined by 8 cubes, although most of them are distorted due to the geometrical projection (in the same way that most squares in the geometrical projection of a cube on a plane are distorted due to the geometrical projection.)

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#15

Re: Dali and the Fourth Dimension: "Impossible" Objects, Part II

03/12/2013 10:27 AM

A 'rotating' tesseract...

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#16

Re: Dali and the Fourth Dimension: "Impossible" Objects, Part II

03/12/2013 10:56 AM

Yes, I have already seen such animations in the past. It's fun to "chose" one of the cubes and watch it changing its position and be distorted, as the Hyper-cube is rotating.

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#6

Re: Dali and the Fourth Dimension: "Impossible" Objects, Part II

03/09/2013 12:46 PM
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#7

Re: Dali and the Fourth Dimension: "Impossible" Objects, Part II

03/09/2013 1:04 PM
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#9

Re: Dali and the Fourth Dimension: "Impossible" Objects, Part II

03/09/2013 4:02 PM

#6 didn't show up on my page... here is a different version.

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