Revisit our Last Meeting

I’d like to get a sense of how you are all feeling about these series of investigations into space and perception. Please complete this form.

Before starting our investigation into capturing and representing space, let’s take another look at patterns created by light and fractals. We didn’t have a lot of time when we last met to explore some of these phenomenon.

4.3 Patterns (transitional objects, fractal dimensions)

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The cases below can help us better understand the complex nature of space, dimension, and perception. Fractals, for example, are said to behave both like a line and a plane, giving them dimension designations like 1.5435 or 1.9. This ratio between first and second dimensions reflects the complexity of the line in space and usually approaches 2. While the more simple the nature of the line, the more the fractal behaves like a one-dimensional line. The study of topology can help us further understand the ways mathematicians describe surfaces and the phenomenon of spaces with holes in them. Although we already saw Powers of Ten and an example from the Storm King exhibit, both of these cases help us consider perspective mediated by distance and light. Let’s take some time consider the cases below.

5.1 Philosophy Time

Freudenthal was a mathematics educator/researcher that felt mathematics should be a human activity of reinvention. He developed Realistic Mathematics Education (RME) with the goal of helping learners grasp everyday reality. For Freudenthal, this reality should be the fundamental starting point of mathematics instruction. Do you feel your ability to “grasp everyday reality” has changed at all since we started investigating space and perspective?

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Let’s revisit Merleau-Ponty  and Freudenthal by reading a few of their ideas:

  • “The world of perception…revealed to us by our senses and in everyday life, seems at first sight to be the one we know best of all…[y]et this is a delusion…[it is] unknown territory as long as we remain in the practical or utilitarian attitude” (Merleau-Ponty, 1948/2004, p. 31).
  • “And since it is about the education of children, [geometry] is grasping that space in which the child lives, breathes and moves. The space the child must learn to know, explore, conquer, in order to live, breath and move better in it. Are we so accustomed to this space that we cannot imagine how important it is for us and for those we are educating?” (Freudenthal,1973, p. 403).
  • Merleau-Ponty (1948/2004) concerning Euclidean geometry: “we have a world in which objects cannot be considered to be entirely self-identical, one in which it seems as though form and content are mixed, the boundary between them blurred. Such a world lacks the rigid framework once provided by the uniform space of Euclid. We can no longer draw an absolute distinction between and the things which occupy it, not indeed between the pure idea of space and the concrete spectacle it presents to our senses” (p. 39).


Don Ihde is a philosopher at Stonybrook and writes about the intersection of technology and culture, or what he terms technoculture. He uses abundant examples from history, art, and science, and theories from other disciplines to demonstrate the way perception is inextricable linked to time, space, culture, and the self. For example, his study of art depicted from illiterate and literate cultures shows that literate cultures tend to create a perspective from the point of view of a reader, a sort of angled birds-eye view of landscape (see Fig. 1). Alternately, illiterate cultures tend to create depictions that are horizontal in nature with a different relationship between the figures. In such artworks, for example, birds and humans are often depicted in various orientations, seemingly unrelated to the landscape (see Fig. 2). Most interesting is art from ancient Egypt, where multiple perspectives are portrayed in a single work (see Fig. 3). How can we attend to these various perspectives as a source for investigating space? Let’s take a moment and jot our ideas down before we talk.


Figure 1. Bird’s eye view


Figure 2. Horizontal view.


Figure 3. Multiple perspectives.

5.2 Discuss Flatland, Part I, Sections 8-12


Last time we met, I asked you guys to continue reading Flatland, Part I, Sections 8-12. These particular sections continue to describe Flatland and activities such as painting, the color bill, chromatic sedition, and the circle priests. I’d love to know your thoughts on these sections, especially the idea of painting a 2D creature who can only see in 1D. Really! What does that (color) look like with no height? Let’s talk.

Capturing & Representing Space

5.3 Mediating Technologies

galileo tele

Space (in an astrological sense) and all the objects occupying it have guided many explorers and farmers since the beginning of mankind. The Earth is so large in comparison to us (remember Space, The Sun, The Earth, and Us). The relative size of the Earth and the way the sun appears to travel east to west could cause the conclusion that the Earth is at the center with an orbiting sun. However, Galileo was able to show that the sun is actually at the center of our solar system by using a mediating technology, the telescope. Let’s take a look at some mediating technologies and come up with other technologies that may mediate our perception of space.

5.4 Lenses and Filters


Lenses and filters alter the way we both capture and see objects in space. Some lenses, such as a fisheye lens actually distorts objects – both hiding and revealing. Let’s explore some cases with lenses and filters. If we have time, we should experiment with apps and cameras outside to see what we can capture for each other.

5.5 Light


Light infiltrates our space and gives us the ability to see objects. However, light is directional (meaning it shines from a source). In the photo of the Austin Mini Copper above, you can see how taking a photo into the light actually captures the direction of the suns rays. The sun was setting at this point (on the west side of the mini). The cases below examine light with a collection of great stories from the Nikon camera site. In addition, more photos of the sun from various angles are shown in Pat/Simon and Sun.

5.6 Art, Photography, Videography Techniques

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Upon first meeting my husband, Pat, I knew that I would need to enjoy hockey if the relationship was going to work – becoming a Red Wings fan the year they won the Stanley Cup made this possible. What I didn’t foresee was the amount of racing we’d be watching together (motorcycle, car, dirt track, demolition derby – you name it). What I’ve realized in being a spectator (live and on TV) is that techniques for capturing the motion of a game or race have been advancing rapidly. The 2013 Super Bowl game was inundated with announcements for the “fan cam” – did any of you watch this? Below are some cases in the art, photography, and videography world where techniques are employed to help you see more (and less) – from multiple angles – and even adjustments to speed.

5.7 Games

Super Mario Brothers

Super Mario Brothers

Have any of you played Mario Brothers? You know how the worlds/levels are denoted by 1-1, 1-2, 1-3, 2-1, 2-2, 2-3, etc. Let’s take some time to contemplate a Mario Brothers of world 1-1/line creature in lineland, world 3-4/space creature in hyperland), etc. Would this be possible to design? Let’s take a moment now and sketch out some possible ideas. Feel free to use Power Point or a pad of paper to help. If you guys are up for it, we could form teams and each take on a particular world/level.

5.8 Photographing Space Activity


Last time we met, I asked you all to bring cameras and even filters or apps you’d like to try out. I can’t foresee the weather while writing this, but either way, let’s take some time to walk around the school experimenting with our cameras. Try to play around with settings, zoom, speed, aperture, filters – anything. Let’s plan to meet up in 20 minutes. At that time, we can upload these to our blog sites and post them for everyone to view.




Flatland, Part II, Sections 13-17: These sections describe A. Square’s dream of Lineland and also introduces a visitor from the third dimension. Think about how the book is similar to and/or different from Flatland: The Movie.

Photographing Space Follow-Up

Finish uploading your photos to your blog site. You may add more later if you’d like. It would be cool to have a page dedicated to your photos – maybe this is a start. Over the next week, write a reflection woven with your photos. Be as creative as you like and be sure to check out your classmates’ pages. This would be a great time to practice creating digital communities and extending your social presence beyond Facebook or Twitter. Blogs are a great way to collect your thoughts and ideas, even displaying your creative acts with others. It would be wonderful to continue commenting on classmates postings that you find engaging – let them know what you think.

Blog Prompt (choose):

  • Take photographs of an object from multiple points of view and post these on your blog. What varies/stays the same in your photos? Is there a point of view you wanted to capture that you couldn’t – describe?
  • Search for skateboard photos that use a fisheye lens. Describe your perspective of the skater and landscape.
  • Search for Internet photos of objects in space (moon, planets, stars). How is this image different from what you are able to see without the lens? What is your natural sight able to capture that the photo doesn’t?
  • What happens if you roll the surface of a tic-tac-toe game into a tube (torus)? How does the game change? This “torus” effect is evident in older video games, like Asteroids. Look this game up and write about how this is similar or different from the rolled up tic-tac-toe game.
  • What sorts of perspectives do video games allow? Try looking up some older games and reflecting on more modern games you’ve played.

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