Shape 

Copyright David & Cynthia Thomas, 2009

 

Visualizing 3-Dimensional Objects--Concepts and Connections

 

Challenge: What Can You Build With 36 Cubes?

 

The possible rectangular boxes (i.e., right rectangular prisms) are those for which length x width x height = 36.   Every permutation (i.e., ordering) of the same three factors could be thought of as representing a different box.  In this case the products 1x2x18, 2x1x18, 1x18x2, and so on would represent different boxes.  How many different permutations of three factors are possible?   What relationships exist between the boxes associated with these permutations? 

 

A different approach would be to think of different permutations of the same three factors as representing the same box.  For instance, the products 1x1x36, 1x36x1, and 36x1x1 could be thought of as representing the same box.  In this approach, how many different boxes are possible?  What relationships exist between the boxes?  Why do all of the boxes have the same volume?  Which box has the maximum (exposed) surface area?  Why?  Which box has the minimum surface area?  Why?

 

Directed Activities Sketching Isometric Views and Sketching Front-Top-Side Views.  A critical visualization skill is that of imaging how a 3-D object would look as seen from different points of view, or perspectives.  The NCTM Illuminations applet Isometric Drawing Tool provides a convenient and powerful tool for exploring alternative points of view.  For instance, the isometric view in Table 6.2 shows an arrangement of 6 cubes.  Three other representations present the object as seen from the front, top, and right side.  Whether using actual blocks or a computer modeling tool, students should imagine what a given arrangement of blocks would look like from these perspectives, make a quick sketch representing what they have imagined, then check their thinking by positioning themselves or manipulating the computer model accordingly.   The NCTM i-Math Investigation Spatial Reasoning Using Cubes and Isometric Drawings provides an introduction to this topic.

 

Isometric View

 

 

 

 

Front View

 

Top View

 

 

Right Side View

Table 6.2: Alternative Views of 6 Blocks

            A related skill involves creating an isometric view of an object given its front, top, and side views.  Given the views presented in Table 6.3, sketch an isometric view of the object.  For some students, this sort of visual analysis and manipulation is easy.  For other, it is frustrating and confusing.  As with most thinking skills, spatial visualization may be learned and mastered with concentration and practice. 

 

Top View

 

 

Front View

 

Right View

 

 

 

 

 

 

Table 6.3: Constructing an Isometric View from Top, Front, and Right Side Views

 

Directed Activity The Concept of Dimensionality.  How is a point different from a line? How is a line different from a plane?  How is a plane different from 3-dimensional space?   Because points, lines, planes, and 3-D space are mathematical abstractions rather than real world objects, their attributes and relationships must be deduced using logic.  Nevertheless, real world objects and their graphical representations may be used to develop concepts and discover relationships.  Table 6.4 demonstrates this use of graphical representations and the mathematical concepts they represent.

 

Figure

Name

Dimension

Description

Point

0

 

·        A point is nothing more than a location or position.  Consequently, points have neither width nor breadth. 

Segment

1

·        As point A moves along a straight line, its successive locations determine a line segment.  Consequently, line segments have length but no breadth. 

Rectangle

2

·        As a segment AB moves in a direction perpendicular to itself, its successive locations determine a filled-in rectangle.  Consequently, rectangles have both length and breadth.

Box

3

·        As a rectangle ABCD moves in a direction perpendicular to itself, its successive locations determine a filled-in parallelepiped, or box.  Consequently, boxes have length, breadth, and depth.

Table 6.4: Representing Objects of Different Dimensions

 

            As suggested by Table 6.4, the dimensionality of an object is related to the sort of space that it occupies.  For instance, a 1-dimensional space such as a line m could contain many line segments, each having its own length.  Assuming that no segment could pass thorough another, each segment would have exactly two “neighbors” on the line, one to its right and the other to its left (See Figure 6.8).  If line segments such as AB, CD, and EG were conscious entities, what would their space “look like” to them?  What sorts of interactions would be possible between such beings?

Figure 6.8: Objects in a 1-D Space

 

            A more familiar situation is presented in Figure 6.9.  As students of geometry, you are accustomed to examining the features of 2-dimensional objects and their relationships.  But what if the polygons were conscious entities in a 2-dimensional space, free to move about and interact with one another?  What would their space “look like” to them?  And how might the beings of this 2-dimensonal space regard the 1-dimensional creatures depicted in Figure 6.8?

 

Figure 6.9: Objects in 2-D Space

 

In the book Flatland, Edwin Abbot, a nineteenth century school master, used this sort of encounter as a metaphor for examining the concept of dimensionality.  After reading Flatland, you might well wonder what sort of difficulties you, a 3-dimensional being, would experience if you ever encountered a 4-dimensional being.  One of the most widely read mathematical stories every told, Flatland is as popular and valuable today as it was when it was first published in 1884. 

 

Directed Activity Geometric Solids & Euler’s Formula.  A polyhedron is a 3-dimensional object bounded by polygons, or faces.  Given any polyhedron, the sides of these polygons comprise its edges and the vertices of the polygons comprise its vertices.  If the faces of a polyhedron are all identical, then its edges and angles are also identical.  Polyhedra with this characteristic are said to be regular.  The complete set of regular polyhedra is known as the Platonic Solids.  An excellent tool for exploring these objects is the Platonic Solids applet at the National Library of Virtual Manipulatives.  Using this applet, you may size, rotate, and color the polyhedra or view them as wire frame figures (See Table 6.5).  The applet also lists each polyhedron’s faces, edges, and vertices. 

 

 

4 faces, 6 edges, 4 vertices

Tetrahedron

 

6 faces, 12 edges, 8 vertices

Cube

 

8 faces, 12 edges, 6 vertices

Octahedron

 

12 faces, 30 edges, 20 vertices

Dodecahedron

 

20 faces, 30 edges, 12 vertices

Icosahedron

 

 

Table 6.5: Platonic Solids

 

When the Platonic Solids’ faces, edges, and vertices data are arranged in tabular format (See Table 6.6), an interesting relationship emerges.  Note that, in each case, the number of faces plus the number of vertices is two less than the number of edges, or F + V = E – 2.  This relationship is known as Euler’s formula.

 

Object

Faces

Vertices

Edges

Tetrahedron

4

4

6

Cube

6

8

12

Octahedron

8

6

12

Dodecahedron

12

20

30

Icosahedron

20

12

30

Table 6.6: Faces, Vertices, and Edges for the Platonic Solids

 

Table 6.7 contains faces, vertices, and edges data from the set of semi-regular solids known as the Archimedian polyhedraThe faces of these 13 convex polyhedra are comprised of two or more different types of regular polygons arranged in the same way about each vertex with all sides the same length.  Check the data in Table 6.7 using Euler’s formula.  Does Euler’s formula apply to other polyhedra?   To all polyhedra? 

 

Polyhedron

faces

vertices

edges

cuboctahedron

14

12

24

great rhombicosidodecahedron

62

120

180

great rhombicuboctahedron

26

48

72

icosidodecahedron

32

30

60

small rhombicosidodecahedron

62

60

120

small rhombicuboctahedron

26

24

48

snub cube

38

24

60

snub dodecahedron

92

60

150

truncated cube

14

24

36

truncated dodecahedron

32

60

90

truncated icosahedron

32

60

90

truncated octahedron

14

24

36

truncated tetrahedron

8

12

18

Table 6.7: Faces, Vertices, and Edges for the Archimedian Solids

 

Directed Activity Slicing Platonic Solids.  Visualizing the intersection of a plane and a 3-dimensional object can be a challenging task.  If you have access to 3-dimensional models of the Platonic Solids, you may find the following hands-on approaches helpful.

·        Dip an unpainted wooden Platonic Solid partway into a container of water, taking care not to change the model’s orientation relative to the surface of the water.  When the model is removed, the wet portion of the model will be darker than the dry portion.  Examine the waterline. What 2-dimensional intersection does the waterline suggest?

·        Use a rubber band on the outside of a Platonic Solid to approximate the intersection of a plane with the surface of the model.  What considerations govern the positioning of the rubber band? What 2-dimensional intersection does the rubber band suggest?

Partially fill a transparent Platonic Solid with water.  As the model is rotated, the surface of the water defines different intersections.  How does changing the amount of water inside the model change the intersections?