# Continuous Extensions

*a priori*with him or herself as a subject existing in this universe. The primitive universe is discovered through action in progress. Piaget explains this progress through subsequent stages of what he calls sensorimotor functions, or, combinations of sensory and motor actions. His studies challenge the idea that a child is born with an innate geometric intuition of the universe through which he or she understands an object’s metric properties independent of his or her actions upon it.

Piaget’s studies of perceptual development show a child organizing space through primitive notions of topology before comprehending higher degrees of geometry and spatial coordinates. At first, Piaget says, the world is experienced as elastic. Early objects are one-dimensional, morphing into new forms when turning or moving in space rather than expressing new perspectives of stable objects. Space is initially perceived as spherical, similar to the way ancient civilizations viewed the outer world as a sphere of images. In a spherical space, stars and constellations are seen as forms sitting on the surface of a sphere, rather than as objects in distant space. Volumes and masses in this perception of space are flattened and seem to merge as they overlap in front or behind one another, just as shadows merge into one shape. With the aid of prehension, grasping and seizing objects, the sphere of space recedes and a child begins to develop the sense of shapes and solidity—a consistency not relatable in earlier stages when objects seen at a distance were perceived as flexible and elastic. By the age of two, a child may coordinate the relationships of objects together, the reversal of objects, their rotations and positions. After days and months touching and observing objects and their displacements, a practical understanding that objects are “free,” existing in the world in their own right and independent of the child’s action on them, slowly starts to develop.

It is at this point that a child’s world is no longer primarily phenomenal, for he or she can now comprehend the “how” of appearance.This transition occurs with the ability to imagine a vanished object as a mental image. With this last stage a child develops a belief in permanency in objects, even those not visible to immediate perception. Not only does a child understand an object as existing outside his or her observation (touch, smell, sense), but the body itself is now also regarded as an object. Space has evolved from perceptual activity to become partly representational.“...the object is now definitely freed from perception and action alike and obeys entirely autonomous laws of displacement. In effect, by virtue of the very fact that it enters the system of abstract or indirect images and relations, the object acquires in the subject’s consciousness, a new and final degree of freedom.” Piaget

See for instance my niece Ada's representational depiction of me: Ada Sillings (5 years old),

*aunt Becca*, (2015) and her sister Caroline's early scribbling: Caroline Sillings (3 years old),

*untitled*, (2015).

In their description of a child’s conception of space, Jean Piaget and Barbel Elizabeth Inhelder put forward the idea that human intuition is shaped through a succession of constructed knowledge. In their studies from the 1950s, Piaget and Inhelder concluded that an individual’s development of spatial intelligence could be described through the hierarchy of geometry—in the order of the topological, projective, affine and Euclidean-metric. It would be too simple, they said, to assume that the representation of space, a process situated at the perceptual level and the level of thought and imagination, copies the existing sensorimotor constructs. The sensorimotor stages develop through a constructive process unique to its own criteria for discovery. At the point when a child begins drawing, he or she does not begin with symbolic imagery or spatial representation, despite the fact that, perceptually, objects and subjects exist outside actions performed on them. Drawing appears to ignore metric and perspectival properties. “Consequentially, it is forced to reconstruct space from the most primitive notions such as the topological relationships of proximity, separation, order, enclosure, etc., applying them to the metric and projective figures yielded by perception at a higher level than that of these primitive relationships themselves.”

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Just as in sensorimotor construction, primary notions of topology—proximity, order, separation, and enclosure—characterize a child’s earliest illustrations. Before the age of two, the spontaneous drawings of children are characterized by scribbles, a primary feature which is an expression of rhythm—either as a forward and backward, or circular, motion of the hand across the page. Piaget and Inhelder relate these gestures to the operational ideo-motor activity from whence later accurate drawings of shapes and geometries will emerge. The construction of scribbles with rhythmic straight and curved lines presupposes a child’s later ability to represent curves, straight lines, and angles of shapes. Primitive notions of topology will visually be seen to progress over the next few years, between ages of two and six. The abstraction of lines and curves from scribbles will be arranged as elements to reproduce geometric shapes, and pictorial representations organized by topological elements abstracted from the scribbles will be used to form a likeness of an image being imitated. By the time a child reaches the age of six, his or her drawn shapes and forms will be correctly reproduced, though projective representations—such as depicting lines of tables that angle in distance—continue to prove difficult until the age of seven. By the age of six a child may start representing an object’s rotations and movements, but these reproductions of objects still seem to interpret the object as something which exists because of the actions performed on it. Representation at this stage is subjective, and in this projective viewpoint, an object is depicted as a co-ordination of elements relative to the point of view of a child. Between seven and eight, a child will begin to represent perspectival space, leading to the construction of coordinate system through which he or she may be able to define the volume of an object in the Euclidean plane through sets of points that convincingly express distance and angles. Through the metrics of Euclidean geometry, an image will eventually be represented objectively.

Inhelder and Piaget defined the transition from projective to Euclidean space as a transition through affine translations. To demonstrate this transitional moment in image making, children were shown a drawing of a Lazy Tongs, closed, and asked to predict, through a series of drawings, the transformation of spaces when the tongs were opened. Those children correctly anticipating affinitive changes of the “windows” delineated in the central negative spaces of the apparatus, not only represented the shapes enlarging then narrowing when the scissors opened and expanded, but also maintained the correct measurement for each side of the transforming shape.The correctness of these representations may indicate that it is no longer regarded as an assimilation of constructional schemata as in earlier stages of representation, but as elements distinct yet connected to a coherent whole—a coordination of parts held together by a stable reference point. At this time a child starts to develop a sense of the object

*itself*through the translations of the

*object-in-itself*: the translations no longer constituted by topological relations and a constructional schemata, but through the measurements and properties of the object being observed. The depiction of displacement points in affine transformations indicate various vanishing points for an object. These are optional viewpoints, a freedom of representation outside a subjectively constructed world and not yet bound to the three-dimensional plane of Cartesian space. At this stage a child has developed an internal system of coordinates for images, exhibiting an intellect capable of grasping a logic of measurement, parallelism, proportion and object displacements, though not yet capable of imitating correct perspectival coordinate of Euclidean space.

Piaget and Inhelder conclude that, during affine transition, a child begins to shift from an individualistic perspective to a more universal one. This latter stage is important to imitation in representation. True representation, Piaget says, begins when a child may deduce from all possible displacements the position of a vanished object, when before he or she could only anticipate the position from perceived relations.

^{ }As the intellectual construction of an internal coordinate system sharpens, a child also comes to recognize him or herself as an independent subject. From this emerges the representational awareness of the viewpoints of others. It is at this age, Piaget and Inhelder say, that the child has freed his or her thoughts from an egocentric point of view. Only now, according to Piaget, will a child be able to predict the form that a shadow of an object may take with respect to a lamp.