chapter 4:

Three-Dimensional Drawing

"Three-Dimensional Drawing" or simply "3-D drawing" is a type of drawing that shows an object as solid, rather than flat. The best way to understanding this idea is to look at a "2-D drawing" that is "flat" and compare it with a "3-D drawing" that appears to be solid. The drawing on the left below is a drawing of the top view of a shoe-box, and the drawing on the right is a 3-D drawing of the same box. Notice that the top view is just a flat rectangle, while the 3-D drawing shows more than one side of the box at a time, and appears more realistic:

In the 2-dimensional view, we see only the top, while in the 3-D view, we see top, front and right side. This is true of all type of 3D drawings. But before we go further with 3-D drawings, we need to clarify some geometric terms, called Geometric Solids.

Geometric Solids

You are probably already familiar with the basic geometric solids as shown below:

We are all familiar with the prisms, the simplest and most well-known geometric solid of all! This particular example is a multifamily housing complex, designed by the architect Moshe Safdie and built in Montreal, Canada. The structure is composed of stacked modular units, all of which are rectangular prisms.

The photo above, and the ones that follow, came from a fascinating website about architecture:

http://www.GreatBuildings.com/

Test Question #1: How is the Great Buildings Online website described, on the opening page? Answer clearly and completely, in at least 20 words.

You can find examples of geometric solids in architecture throughout the world. Of course we all have heard of the Great Pyramid of Khufu, El Giza, Egypt, created in approximately 2500 BC:

The Guggenheim Museum in New York, designed by Frank Lloyd Wright, appears to be in the shape of stacked cylinders; each cylinder is smaller at the bottom than the top:

The geometric solids are also the basis of many complex forms, in art and in architecture. You will see many varieties of geometric solids, both simple and complex, in buildings. For example, look at the curved geometric shapes in Le Corbusier's beautiful Church called Notre Dame du Haut, in Ronchamp France:

. . . and the fascinating 3-dimensional geometry of the Sydney Opera House in Australia:

Some modern buildings are pure, geometric abstract shapes, such as the Vitra Design Museum by Frank Gehry, in Weil-am-Rhein, Germany, built in 1990. This building is a work of art, made up of prisms, cylinders and a helix:

The interiors of buildings, and the furnishings also are geometric forms. The "helix" is a graceful three-dimensional geometric form; you may be most familiar with it as a "spiral" stair:

In art, we see many examples of these same geometric forms in sculpture and in painting. Many artists begin a still life by sketching the geometric shapes that are the underlying forms in the painting, as you can see in the two sketches below. Artists very often begin a painting by sketching the basic geometric and freeform shapes, as you see. Notice the prisms, cylinders in the painting below, and also circles, ellipse and other 2-Dimensional geometric shapes. This artwork is from the web site of the National Gallery of Art:

http://www.nga.gov/home.htm

Test Question #2: As written in the Site Map page at the National Gallery of Art web site, what is the mission of the National Gallery of Art?

In the process of creating a painting, even the human form is often broken down into basic geometric shapes: cylinders, cones, spheres and prisms. When the artist is laying out the general idea for the painting, he or she often draws a "stick figure" or "maque" as seem in the drawing below, from an interesting web page on how to draw:

In this very helpful guide to drawing the human form, the author says "Think of the figure as a series of solid geometric forms which can move in relation to one another. Before developing detailed mass, an artist must learn to think of the figure as a series of solid volumes which move in relation to one another. Cubes, spheres, cylinders, and triangles can be used to depict the figure geometrically." After this preliminary layout, the artist would smooth out the shapes, soften the contours, and in the end create a natural looking human form.

Many pieces of art, from paintings to sculptures, are themselves entirely geometric. These photographs are also from The National Gallery of Art: http://www.nga.gov

Geometric figures can, themselves, look like a work of art, such as the beautiful 3D geometric figures , called polyhedra, shown below. " A polyhedron is a three-dimensional solid whose faces are polygons joined at their edges." (The plural of polyhedron is polyhedra.)

http://mathforum.com/alejandre/applet.polyhedra.html

You can see an amazing origami piece by going to the following web pages, folded from a single dollar bill.The first web page is the first in the step in showing you how you the steps in folding: http://origami.kvi.nl/photos/fit-joe. To see the finished origami: http://origami.kvi.nl/photos/fit-joe/finished.jpg

George Hart is an artist who describes himself as ..." a sculptor of constructive geometric forms, my work deals with patterns and relationships derived from classical ideals of balance and symmetry.  Mathematical yet organic, these abstract forms invite the viewer to partake of the geometric aesthetic.

Leonardo Project

Visit his website, and you will see some fascinating mathematical scuptures: http://www.georgehart.com/sculpture/sculpture.html

Click on the link http://www.georgehart.com/virtual-polyhedra/vp.html and you will find definitions and examples of these mathematical shapes.

Now that you have learned how Art and 3D Geometry are related, you are ready to learn how to do your own 3-Dimensional drawings.

The following link will take you one of my own sets of web pages, that I created a few years ago as part of an online summer workshop at the Math Forum. The Math Forum is at Swarthmore College in Pennsylvania, and I attended this workshop online, sitting at my desk at home in Honolulu! This is just one of many examples of how great Distance Learning and the internet can be.

Because I created this set of web pages specifically to teach geometry students how to create 3-dimensional drawings, and to show them the connections between 3D drawing and mathematics, I would like you to read all of these web pages, not just the first one. The last page of this set describes a project, and that project is what you can do to create your own MathArt Connections project!

http://forum.swarthmore.edu/workshops/sum98/participants/sanders

Linear Perspective is a mathematical system for creating the illusion of space and distance on a flat surface such as a canvas or wall. Perspective drawing has a long and fascinating history. The system originated in Florence, Italy in the early 1400s. Leonardo Da Vinci was one of the greatest painters of the Italian Renaissance, and was very interested in the study of perspective. The following web pages will tell you many fascinating things about Leonardo, and about perspective drawing and painting. You will find many links from these pages, which you are welcome to explore if you like; there are interesting information and activities on those other pages. You might want to begin with the four pages linked below:

http://www.mos.org/sln/Leonardo/LeoHomePage.html

http://www.mos.org/sln/Leonardo/LeonardosPerspective.html

http://www.mos.org/sln/Leonardo/LookingThroughLeosEyes.html

http://www.mos.org/sln/Leonardo/LeonardosWindow.html

In the painting on the left below, you see a road going off into the distance. This is a one-point perspective, and in the second image I have drawn the lines of perspective, all receding to the one vanishing point:

(from http://www.sanford-artedventures.com)

In May of 2002, the Getty Museum in Los Angeles created an exhibition called The Geometry of Seeing, about the connections between perspective, art and mathematics. You can see part of this fascinating exhibit online at http://www.getty.edu/art/exhibitions/geometry/.

You will find examples of perspective everywhere around you, because this is the way our eyes see the world. Wherever we look at the world around us, we see perspective. The following are just a few examples of One- and Two-point perspective. The photos are from http://www.GreatBuildings.com/gbc.html

One point perspective: Stoa of Attalus, Athens, Greece, 150 B.C.

Two Point Perspective: Johnson Wax Building by Frank Lloyd Wright

Chapter 4 Project

The project for this topic is to create an artistic 3-D drawing of a castle. First do some research on the Internet, and find some other images of castles. Read some information about the castles, and explore any aspects that interest you. Then design and create a 3-D drawing of your own castle, using geometric solids. You should have at least one each of the four solids we have studied: prism, pyramid, cylinder, cone. If you would like to see some examples of graphics created by students like yourselves, please go to the Student Work section from the Table of Contents.

• You should use only geometric shapes in your design (prisms, cylinders, pyramids, cones etc.)
• Draw the graphic using compass and ruler, in pencil on unlined white paper, then go over it in pen. Then color it using colored pens or pencils. Or you may construct the graphic using The Geometer's Sketchpad, hide all the points, print it out and then color it with colored pens or pencils. Follow the instructions on the web pages above. (http://forum.swarthmore.edu/workshops/sum98/participants/sanders)
• Write a description of the types of solids that you have used (for example: "the base of the castle, and the walls are prisms with rectangular bases; the towers are prisms with triangular bases and the roofs of the towers are cones..." etc.)

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