The topology of the human form

What is topology?

"Basically, topology is the modern version of geometry, the study of all different sorts of spaces. The thing that distinguishes different kinds of geometry from each other (including topology here as a kind of geometry) is in the kinds of transformations that are allowed before you really consider something changed. (This point of view was first suggested by Felix Klein, a famous German mathematician of the late 1800 and early 1900's.)
In ordinary Euclidean geometry, you can move things around and flip them over, but you can't stretch or bend them. This is called "congruence" in geometry class. Two things are congruent if you can lay one on top of the other in such a way that they exactly match.
In projective geometry, invented during the Renaissance to understand perspective drawing, two things are considered the same if they are both views of the same object. For example, look at a plate on a table from directly above the table, and the plate looks round, like a circle. But walk away a few feet and look at it, and it looks much wider than long, like an ellipse, because of the angle you're at. The ellipse and circle are projectively equivalent.
This is one reason it is hard to learn to draw. The eye and the mind work projectively. They look at this elliptical plate on the table and think it's a circle because they know what happens when you look at things at an angle like that. To learn to draw, you have to learn to draw an ellipse even though your mind is saying `circle' so you can draw what you really see, instead of `what you know it is'.
In topology, any continuous change which can be continuously undone is allowed. So a circle is the same as a triangle or a square because you just `pull on' parts of the circle to make corners and then straighten the sides, to change a circle into a square. Then you just `smooth it out' to turn it back into a circle. These two processes are continuous in the sense that during each of them, nearby points at the start are still nearby at the end." Robert Brunner

More examples of the topology of the human form as mapped with various CAD solutions, i.e, the types of meshes that wrap the geometry. Each kind of mesh has certain parameters that it follows. This is determined by the mathematical algorithm that defines where various coordinates are in space,.For example, a mesh may be perpendicular to the ground plane, or it can be at a designated angle to the ground plane. The mesh is often thought of as being a regular matrix of polygons, projected on or dividing up a volume. The mesh by extension creates a way for the brain to visualize the displacement or occupancy of that volume in space.

Bilateral Symmetry and Congruency

Here are a few more examples of the curious visual phenomena that occur when the inverse surface of the volume ( the inner skin ) flips to look like the obverse or outer skin. This odd mirrored bi-symmetrical flipping occurs through z axis, (into the picture plane) and is illustrated here more clearly by observing the figure with and without hair. Notice that the form flips toward and away along a transverse axis (3/4 point of view.)
This phenomena is commonly referred to as congruency.

Convex or Concave?

Here, the first figure appears to be facing away looking up to the upper right. The lower figure appears to be facing us looking toward the lower right. They are in fact the same figure with and without hair.

From outline to spatial visualization

One visualization strategy is to conceptualize the figure as a mannikin rather than become too enamored of the anatomical details. A generalized approximation of geometry that resembles the masses of the figure emulates how the sculptor builds the volumes. For example, a peanut shape could be analogous with the torso, an ovoid for the head, cylinders for the arms and legs, etc.These primitive geometric forms are more easily mapped in the brain as we shall later find out, and create a volumetric placeholder which is ultimately more accurate than a finely placed contour.

What we are trying to train ourselves to see is the complete form in all its volumetric occupancy.
Illustrated here are CAD renderings of the human figure as groups of polygons that are "skinned" over a volume. From these renderings, it's possible to see more clearly the inner walls of the reverse side of the figure. Also noticeable is how the inner reverse wall exhibits the odd perceptual phenomena of appearing both concave and convex at the same time. Seeing through a wireframe CAD drawing or cross-sectioned rendering allows one to see the bi-symmetrically of the volume very clearly. In this example I sectioned a human figure into two halves along a natural lateral curve, dividing the figure into anterior and posterior halves. Separating the two halves it is possible to see the shell of the hidden back half of the volume.
The last picture illustrates the bi-lateral symmetry in action. I have duplicated the front half and mirrored its symmetry. By comparing the inner shell with the outer shell, it is virtually impossible to tell if the figure is convex or concave.

Convex or Concave???

Origins of Contour Drawing

 The contour drawing defines the perimeter of the figure against its background, as so eloquently illustrated in the Durer engravings circa 1530 .Photographic realism before the camera often relied on the sight size execution as well as camera obscura and various grid transfer techniques. These are all forms of measurement that rely on the projection of a matrix grid that corresponds precisely with the image seen from one point of view, hence the stationary "eye pointer" that establishes the position of the viewer's eyeball.

Mechanics of Sight

To understand the process of visualization, it is important to first consider its mechanics. This may all sound a bit arcane but I can't help resisting the science nerd part of me. Knowing how the brain processes visual information can afford the artist insight into how to craft their creation. As will be shown later, the artist can intentionally orchestrate a trajectory or visual pathway for the viewer to follow by knowing what visual clues have the most impact. Understanding the cognitive process of how the brain visualizes geometry can also be instructive in imparting a sense of volume in one's work. 

How it works:
 When we open our eyes, the available light that is illuminating our world and reflecting off of objects, is instantaneously transduced by the retina through neural ganglion to the thalamus and then to the occipital cortex of the brain. There it is coded and processed into a mental image of that world around us. Although grossly oversimplified, this is basically a description of the mechanics of sight. Incredibly, the coding and processing of sight all occur almost instantaneously, as the brain updates, cross-references, evaluates, remembers, and learns: all within the blink of an eye.
 As our eyes gaze at the world around us, the electromagnetic energy of light permeates back into the inner lining of our eyeballs which is called the retina (1). The retina is lined with specialized cells, which are stratified into layers. The layers toward the back of the retina are arranged radially into a honeycomb – like configuration. and are the photoreceptor cells- the rods and cones. Photoreceptors ate a specialized type of neuron found in the retina that is capable of photo transduction.The great biological importance of photoreceptors is that they convert light(electromagnetic radiation) into signals that can stimulate the biological processes. More specifically, photoreceptor proteins- Rhodopsin and Opsin absorb photons triggering a change in that cells membrane potential.(2) Stacked by the thousands inside these cylindrical like rods and cones, are membrane cells containing the photoreceptor protein Rhodopsin. Rhodopsin reacts with light and undergoes a chemical reaction which is the first in an entire chain of catalyzed reactions.The mechanism that seems to drive all this is the electrical flux of membrane potential- newly created binding sites generated by changes in the Rhodopsin provoke Sodium ions to enter the cell through a gateway in the cell membrane.When too many Sodium ions saturate the cell it results in a resulting in a flood of Glutamate which neutralizes the excessive accumulation of ions.This process results in membrane potentials hyperpolarizing and depolarizing up along the retinal structure from the rods and cones to the bipolar cells which are stimulated to produce a burst of electrical impulses.(3) It is the bipolar cell that can fire an electrical signal to the ganglia neuron which creates an action potential.This now is the encoded signal which can be transmitted from the ganglion,(4) and then routed to the brain through the thalamus.(5)
This is very simply the process of phototransduction.

Light and Neuronal Activity

How rods and Cones respond to light

Volumetric Drawing and the Human Figure

The class: Volumetric Drawing and the Human Figure  , an ongoing class at theWoodstock School of Art, is a drawing course which uses spatial visualization to interpret volumetric form and applies it to drawing the human figure. Although the figure is its emphasis, the course is less a survey of anatomy and more about the psychology of seeing; what Da Vinci refers to as saper vedere : knowing how to see. It is about data transfer and communication, and how the artist can transcribe his or her own expression into a visual reality with which others can engage. These strategies all contribute to the creation of a cognitive model that is then refined with anatomical as well as mechanical data. This cognitive model in its most developed form is a volumetric manikin figure, similar to a holograph, which is quite literally projected from the artist’s mind onto the paper via his or her visualization skills. Equipped with this spatial visualization tool, the student can apply anatomical information as it is learned; the manikin visualization updating itself every time one draws. In this sense, the artist is not copying or measuring the image that is projected on the retina of the eye, but rather is visualizing a cognitive manikin that is a kind of template the artist uses to draw on top.

The learning objectives of this course are focused on providing the student with tools of visualization and ways to then code this information into a cogent and expressive workflow. I find the categorization of this process as a workflow to be particularly appropriate as it implies that a workflow can often adapt itself to a changing environment. Likewise, the artist can learn to adapt their own cognitive approach to their practice to reflect changing parameters as well as evolving goals.. Within the course syllabus are exercises designed to increase capacity for spatial visualization of three dimensional space. As is apparent to most people once they pause to reflect, they can reach for a coffee cup outside of their field of vision and cup their hand to perfectly conform to the geometry of the cup. This is spatial visualization at work- the mind casting a wider net based on small bits of information and projecting a matrix on the field beyond its periphery.