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Hyper-Geometry, Another Level of Deformation 
Essay, A. A. School of Architecture, 2007

1. From Euclidean to Topological to Hyper-geometry

The best way for defining what I call ‘Hyper-geometry’ can be by comparing it with two other types of geometry: Euclidean and Topological Geometry; and ‘deformation’ is the key which differentiates these three types of geometries the best. Briefly speaking, in Euclidean geometry all the valid deformations for the geometry are ‘movement’ and ‘rotation’. In Topological geometry ‘stretching’, ‘pulling’ and ‘pushing’ are also allowed (till the new geometry keeps staying in the same ‘topological set’) (01) to let the geometry ‘grow’ or ‘deform’ more freely. Deformations in Hyper-geometry are topological deformations but ‘scale-based’; this is the main difference between Topological and ‘Hyper- deformations’: Hyper-deformations occur in ‘levels’, according to the scale differentiations.

Fig. 01: A toroid in three dimensions; A classical example of Topological Geometry
Fig. 02: Diagram demonstrating Cell Replication Levels.

Hyper-deformations happen in different ‘levels’ of scale changes. Hyper-geometry starts topologically deforming and consequently resizing till a specific level and by obeying a specific rule; from that point it enters a new level of deformation which can be completely different from the last one; this again continues since a specific moment and then again the geometry enters a new level of deformation, and this can continue infinitely. Lots of deformations in animate nature, considering their growth, are Hyper-deformations which is a higher level of complexity than Euclidean and Topological deformations.

2. Hyper-geometry, geometry of growth in nature

There are lots of examples in the nature which talk about this kind of geometry. Cellular growth is the base one. When a cell grows its surface starts stretching and enlarging (topological deformation); the cell’s size continues changing till a specific point (let’s call ‘reforming point’); when it reaches that point, the cell starts dividing into two new cells (non-topological deformation); then again each of the new parts start changing size till again they reach their reforming points, in which they enter the new level; and then start dividing into new cells, in the same way as the initial cell. (02)

Fig. 03: ‘The leaves alternate as they ascend the stem from the base of the plant leaving a space between leaves of 2 inches and decreasing in distance to ¼ inch toward the bloom. Each leaf simultaneously decreases in size as it climbs the stem. The plants in this meadow have leaves ranging from 1/8 inch to 6 inches in length.’
Fig. 04: Different and not self-similar levels of growth i
n plants.

The deformations in the geometry of most of the plants, is another example of Hyper-deformation. For instance the way that stems in plants give birth to the new spans: a specific ‘span’ in a stem (which always starts from a ‘node’) starts growing by stretching in the first level, but there are always pre-defined rules that define how long this stage can continue; after that this level finishes, the growth enters a new level by giving birth to a new node, from which the second level starts. Actually the areas between the nodes are the different levels of ‘in-time’ growth, each of which talks about a different ‘scale’. This process is not infinite, since in every level, the spans lose their ability for growth (for example by reducing the size of the growth in each level and reducing the thickness of the span) and this makes the process to stop in some point (let’s call ‘end point’). This process differentiates every level from its ex-level. (03)

What is interesting about Hyper-deformations is that always new ‘materials’ and volume of form are given birth, through the process of growth.

3. Materialization of Hyper-geometry

The fact that amount of material, ‘vertices’, ‘edges’, and ‘faces’ change in Hyper-geometry is another important difference between Hyper-geometry and topological one. In topological geometry, always deformations happen with a constraint of number of vertices, edges, and faces. But in Hyper-geometry by the changes in the scale (in each level) some materials appear or disappear, considering that the scale is growing or ‘de-growing’. What I call ‘de-growing’ is the decrease in the scale which usually doesn’t happen in the nature, but is easily imaginable in the world of Geometry.

Fig. 05: Hyper-transformation from Hexahedron, into Truncated Cube, Cuboctahedron, Icosahedron, and Octahedron.

In the example of stem’s growth, number of nodes (vertices) and spans (edges and surfaces) are always changing. In the same way, when cells divide, there are always new surfaces appearing. The symmetrical changes in polyhedrons can be another good example. When a Hexahedron (Cube) is changing its scale by transforming into Truncated Cube, then Cuboctahedron, then Icosahedron, and finally Octahedron (or vice versa) in different levels, in each level the number of vertices, edges, and surfaces are changing. Appearance or disappearance of material in Hyper-geometry is a fact that makes it ‘natural’ but meanwhile complex, specifically imagining that how a Hyper-geometry can be ‘materialized’ when for example it is supposed to take form out of the world of animate-nature and out of the world of ‘digital’: in the world of ‘physical’ and inanimate nature.

4. Some facts about Hyper-deformations

Some facts about Hyper-deformations look very interesting. First of all, when the geometry is rescaling by deforming, this deformation takes place in the entire ‘system’, and not only in the last levels. This means that, for example when stem is growing in one level (span), it does not mean that only that span is growing toward its ‘reforming point’, but the whole system in all its levels is deforming, but of course in different scales in each level. In this way, all the levels are always ‘live’ and contribute in the system’s deformations according to their level and the scale of the system.

The other fact about Hyper-geometry is that the geometries of different levels do not have to be ‘self-similar’. The examples of self-similar levels of deformations are ‘fractals geometry’ and stem growth, and the examples of fully differentiated levels of growth can be the sweet-potato’s root producing new plants (04), or green plants coming into flowers after a certain level of growth. This is what I personally like about Hyper-deformations: they are self-similar till some levels of growth, but they do not constrain themselves to it; they can fully change when they need to. This algorithm can have very nice urban interpretations.

Fig. 06: A sweet-potato root producing new plants.

There is also another interesting fact about Hyper-deformation and it is the reason of growth, scale-changing, and deformation: we usually expect every phenomenon to change because of ‘forces’ applied on it, being ‘external’ or ‘internal’. The interesting point about natural Hyper-deformations is that they usually start to deform because of the internal forces but deform in response and in ‘adaptation’ to external forces. The good examples can be the growth of sunflower which grows not because of the sun but always toward the sun, (05) or the growth of roots in the sand which is always directed in adaptation with its environmental circumstances.

5. Notes

a. Text

1. refer to “Topology.” Wikipedia. http://en.wikipedia.org/wiki/Topology.

2. refer to “The Cell Cycle.” Cells Alive! <http://www.cellsalive.com/cell_cycle.htm>. and “Cellular Replicators – An Exponentialist View.” http://members.optusnet.com.au/exponentialist/Cells.htm.

3. refer to “White Yarrow: Achillea Millefolium.” Flower Essence Society. http://www.flowersociety.org/Yarrow_plant_study.htm.

4. refer to “The first Book of Farming.” Project Guttenberg. http://www.gutenberg.org/files/16900/16900-h/16900-h.htm.

5. “Sunflower Production - A Concise Guide.” Agriculture & Environmental affairs. http://agriculture.kzntl.gov.za/portal/Publications/LooknDo/SunflowerProduction/tabid/134/Default.aspx>.22. Archigram, Peter Cook, 1972, P: 54

b. Figure

Fig. 01: http://en.wikipedia.org/wiki/Topology.

Fig. 02: http://members.optusnet.com.au/exponentialist/Cells.htm.

Fig. 03: http://www.flowersociety.org/Yarrow_plant_study.htm.

Fig. 04: http://www.flowersociety.org/Yarrow_plant_study.htm.

Fig. 05: A project done in DRL, Phase-I, winter 2007.

Fig. 06: http://www.gutenberg.org/files/16900/16900-h/16900-h.htm.

6. References

1. “Topology.” Wikipedia. 23 April 2007 http://en.wikipedia.org/wiki/Topology.

2. “The Cell Cycle.” Cells Alive! 23 April 2007 http://www.cellsalive.com/cell_cycle.htm.

3. Coutts, David. “Cellular Replicators – An Exponentialist View.” 23 April 2007 http://members.optusnet.com.au/exponentialist/Cells.htm.

4. Ellen, Jane. “White Yarrow: Achillea Millefolium.” Flower Essence Society. 23 April 2007 http://www.flowersociety.org/Yarrow_plant_study.htm.

5. Goodrich, Charles L. “The first Book of Farming.” Project Guttenberg. 23 April 2007 http://www.gutenberg.org/files/16900/16900-h/16900-h.htm.

6. “Sunflower Production - A Concise Guide.” Agriculture & Environmental affairs. 23 April 2007 http://agriculture.kzntl.gov.za/portal/Publications/LooknDo/SunflowerProduction/tabid/134/Default.aspx.

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