Beauty and Truth: Geometry, Perspective and Proportion in the Seventh Grade

Introduction: During the fall of 2017, Karl Johnson guest taught two blocks at the Wasatch Waldorf Charter School in Salt Lake City, Utah. Being one of the newest schools in the States, the students did not have a thorough grounding in subjects which Waldorf students would have normally covered, so he led into Perspective Drawing through Geometry, Geometrical Drawing and the history of the development of Geometry. Karl has served the Wasatch Charter School since their opening in fall of 2016 as a mentor and guest teacher and serves on the faculty of the Gradalis Waldorf Teacher Training program – a training program through which several of their teachers are gaining their Waldorf Teacher certification. While he was there, Karl offered this piece for their newsletter. 

How do we awaken adolescent students to a feeling for goodness and beauty as they begin to unfold new capacities during this pivotal phase of life? How do we help lead them in their dawning search for truth? The Seventh Grade Phoenix Class of the Wasatch Charter School recently completed a remarkable journey through the beauty and “truth” of geometrical forms, through perspective drawing, all the way to the balance and proportion of the form of human being. This journey awoke the students to the innate geometrical sense that we all possess and ultimately tied in history themes from the ancients to the High Renaissance. Throughout there was an emphasis on precision and “lawfulness” even as the students brought forth a great flowering of beauty through their work.

This block began with a consideration of the geometrical forms found in Nature: the straightness of the pine trees, the curve of the wilting flower stalk, the symmetry of the honeycomb, the gesture of the spiral found not only in seashells and pine cones, but also in galaxies and in the growth of hair on the head of a new-born infant. We quickly proceeded into geometrical drawings and a covering of basic geometry topics. The drawings were constructed using the draftsman’s tools of straight-edge, T-square, and compass and then carefully colored. Great emphasis was placed on precision, accuracy and beauty - including the color choices for the shading. The students were able to learn the six-division of the circle, the twelve-division and even the twenty-four- division, and create elaborate forms. The forms were all constructed within a circle, yet within the constructions, a myriad of other forms were discovered: triangles, hexagons, trapezoids, rhombi, rectangles, squares and more.

7th Grade Perspective Drawings - Wasatch Waldorf Charter School

This all led to an examination of the basic elements of Geometry, such as radius, diameters, circumference, point, arc, angles, perpendiculars, intersections, lines, rays, with all the vocabulary and defining characteristics of these various elements.

As we worked, we began a journey through of the history of the development of Geometry - from the “practical” geometry of the ancients to the "thinking" geometry of the Greeks to the "artistic" geometry of the Renaissance arising from the discovery of the laws of linear perspective. The students heard the stories of the beginning of the development of geometry during the ancient civilizations, such as the Mesopotamians’ 360 degree division of the circle in order to map the night sky and the Egyptians’ discovery of the 3,4,5 right-angle triangle as they used knotted
ropes to survey the flood plains of the Nile River each spring.

They also heard the story of Thales, the Greek “Father of Geometry” who accurately calculated the height of the Great Pyramid by using his knowledge of shadows and ratios. We were then able to examine the formula he worked with and this led us to other foundational aspects step-by- step “thinking” geometry and our own ability to “reason” and think logically.

Ultimately, the stories led up to the biography of Filippo Brunelleschi, who in 1413, developed the laws of perspective drawing and opened the new age of the Renaissance. This paved the way for the masterpiece works of that period and for a new awareness of space. The discovery of the laws of perspective eventually helps lead to the development of Projective Geometry.

This fascinating story, from the ancients to the Renaissance, clearly indicates the development of consciousness as thinking progressed through the ages. This is very much akin to what the students in the Seventh Grade year are experiencing themselves as their development unfolds in new dimensions.

During the lessons, the students created a series of plates – first the geometrical drawings and then a series of perspective drawing plates. In the perspective drawings, the students were able to learn the technicalities of vanishing points, converging lines, interpolation and extrapolation and gain the ability to create the illusion of three-dimensional space on a two-dimensional sheet of paper. They also learned about other perspective techniques such as overlapping, atmospheric fading, and shadowing. These perspective plates retained the “lawfulness” of the geometrical drawings, while giving the students even more freedom for artistic expression.

As a class, we also worked up an outline of the history and the students wrote a short paper regarding the history of the development of geometry encapsulating everything mentioned above.

Our story of the development of geometry and perspective drawing led us right into the whole of the Renaissance period. We examined the conditions that led up to this period, when, in Europe, the “world awoke” from the slumber of the previous ages.

Student renderings of Renaissance masterworks - Seventh Grade -Wasatch Waldorf Charter School.

Initially through the biographies of the great trio of High Renaissance artists– Leonardo da Vinci, Raphael, and Michelangelo – we studied this cultural awakening in Florence and Rome during the late 1400’s and early 1500’s. These biographies gave us not only the life story of these interesting figures, but also gave us a flavor of all aspects of the Renaissance – from daily life to the guilds to the sway of the Roman Catholic church during those times to the influence of the Medici family in Florence.

Moreover, through the flourishing of the art of the High Renaissance, we were able to study the beauty and proportion of the human form. With large replicas of masterworks on loan from the Salt Lake City library, we were able to look at the artistry, perspective techniques and symbology of the great artists of the Renaissance –and especially their treatment of the human form. We studied the geometrically balanced proportions of the human face, and were able to sketch several detailed portraits from the works of the masters. We then examined the overall proportions of the whole human form and our journey culminated with renderings of Leonardo’s famous “Vitruvian Man”.

From the geometry of Nature to the symmetry of the human form, these lessons helped the Seventh Grade students gain a feeling for the beauty and truth – of both the world at large and of the human being – as they “awaken” into their own “Renaissance” and step toward the threshold of young adulthood.

Karl David Johnson, MA Mentor/Consultant/Guest Teacher