Lab: Platonic Solids
Activity 1: Constructing a Regular Tetrahedron
Procedures: Open Cabri 3D software, go to “File” > “New From Template…” and select the
“Natural no vectors (US letter)” template. This means we will be doing “synthetic” geometry –
we will not need a coordinate system aside from a unit to measure length.
Part One
1. Construct a large equilateral triangle in the plane given. Select the “equilateral triangle” tool, click on
the plane to construct the triangle on the plane. Select a point near the center of the plane for the
center of the triangle, then place a vertex so that the triangle is relatively large.
2. Construct a line perpendicular to the triangle through the center of the triangle, using the
perpendicular tool.
3. Construct a circle through one vertex of the triangle around its opposite edge using the circle tool,
selecting a vertex, then selecting the opposite edge.
4. Create a new point at the intersection of the circle and the line through the center of the triangle. Hide
the circle and line by selecting each with the “Manipulation” tool and hitting CTRL+M. To view
hidden objects, go to Window > Active View to open the active view window, where you can hide
and unhide objects by selecting “Show Hidden Objects.”
5. Construct a tetrahedron (four-faced polyhedron) whose vertices are those of the triangle and the
intersection point. Using the “Convex Polyhedron” tool, select each of the four vertices exactly once,
then “Validate Polyhedron” by clicking a vertex already selected.
6. Use the Length tool to inspect (hover over) the lengths of various edges. Are they all equal? Does this
guarantee you’ve created a regular tetrahedron, one of the Platonic solids?
7. Using the Manipulate tool, right-click on the tetrahedron. Highlight “Surface Style” and select
“Empty” to see the vertices and edges more fully. Count the Vertices, Edges, and Faces, saving these
numbers for later. You will also need the tetrahedron in the next activity.
Note: Adjusting the slider in the “Active view” window will rotate the workspace, easing
the construction
Activity 2: Finding an Octahedron/Discovering Duals
Part One: Using the regular tetrahedron from Activity 1, construct a related Platonic solid.
Continuing from Activity 1…
1. Having made the surface of the tetrahedron empty, hide the plane, the (green) triangle, and the
midpoint of the triangle (but do not delete).
2. Create midpoints for each of the six edges of the tetrahedron, using the Midpoint tool (fifth button
from left).
3. Create a convex polyhedron whose vertices are the six midpoints, using the Create Convex
Polyhedron tool. Rotating the shape may help. While selecting vertices, hovering over selected
vertices to see the ‘shape in progress’ may also help
4. Use the Length tool to hover over various edges to compare their lengths. Are they all equal? Does
this guarantee you’ve found another Platonic solid?
5. Count the Vertices, Edges, and Faces, saving these numbers for later. You should have created a
figure with 8 faces (half of them coplanar with those of the tetrahedron, half oriented toward each
vertex of the tetrahedron). This is the (regular) octahedron, another Platonic solid.
Part Two: Finding the ‘dual’ of a solid
Continuing from Part One…
1. Using the Manipulate tool, select an edge of the tetrahedron and Hide (CTRL+M) the tetrahedron.
Repeat to hide the ‘top’ point not in the octahedron.
2. Using the Segment tool, create two diameters of the octahedron – lines between opposite vertices.
You may wish to change the surface of the octahedron to empty (as pictured) to better see the
diameters.
3. Create a new point at their intersection then hide the diameters, leaving only the center of the
octahedron.
4. Make the surface solid again (right-click on an edge of the octahedron). Use the Perpendicular tool to
draw four lines, perpendicular to the four pairs of faces, each through the center of the octahedron
(the intersection of the diameters from step 3).
5. Create 8 points, one at each intersection of a perpendicular with a face of the octahedron. (The
bottom is hard to select, but luckily it already exists and is hidden – it is the center of the original
triangle on which the tetrahedron was built. Show hidden objects and make the surface of the
octahedron empty if you have trouble unhiding the centert of the bottom face)
6. Hide the four infinite lines through the center of the octahedron, and make the surface of the
octahedron empty. Construct a Convex Polyhedron whose vertices are the 8 centers of the faces
of the octahedron. This is the dual of the octahedron. Does it look familiar? Assuming the
octahedron you created was regular, is the dual solid guaranteed to be regular?
7. Count the Vertices, Edges, and Faces of the cube, and compare with those of the octahedron.
What do you notice?
8. Having constructed a cube as the dual of an octahedron (left), what can you say about the image
on the right? That is, can you find four special vertices of the cube? How are cubes and tetrahedra
related?
Activity 3: The Last of the Platonic solids
1. Start a new file from template, choose “Natural no vectors (US letter)” again.
2. Using the “Cube” tool, create a medium-sized cube with one face in the plane, roughly centered.
3. Create a line between opposite vertices of the cube, and hide the point at the center of the bottom
face.
4. Create a circle through the three vertices adjacent to the upper vertex of the diagonal of the cube
5. Create another diagonal of the cube, as well as the point of intersection between the two diagonals.
Hide both diagonals.
6. Construct three perpendiculars through the center of the cube (intersection of diagonals)
perpendicular to the faces of the cube using the Perpendicular tool.
7. Create three planes, one for each pair of perpendiculars, using the Plane tool. Each plane will intersect
the circle in two points, for a total of six intersections of the circle and planes. Create a point at each
of the six intersections (you may have to rotate the workspace and hide planes to see all six
intersections).
8. Hide the three planes. The six points on the circle which are not vertices of the cube will form two
equilateral triangles (you do not need to construct them). Hide the three vertices of one of the
triangles.
9. Using the Half-Turn tool, select a perpendicular, and reflect each of the three points on the circle
one by one. Repeat this process for the other two perpendiculars.
10. Hide the three perpendiculars. You should have created 9 points, together with the original three
on the circle, such that two points are ‘above’ each face of the cube. You may wish to make the
surface of the cube empty to verify 12 points not on the cube.
11. Use the Convex Polyhedron tool to construct a polyhedron whose 20 vertices are the 8 of the
cube in addition to the 12 points created. What is unique about this polyhedron compared to the
three previous?
12. Count the Vertices, Edges, and Faces of this shape. After verifying the edge lengths are all the
same, you have constructed the regular dodecahedron, the 12-sided platonic solid.
13. Given the relationship between the dual solids the cube and octahedron, what can you say about
the vertices, edges, and faces of the dual to the dodecahedron? How many are there, and what
polygon is it built from?
EXTRA CREDIT!
In step 8 of the dodecahedron construction you arbitrarily chose one of two triangles. Looking at how the
12 new points are arranged with respect to the cube, make a guess as to whether/how the construction
would have changed had you chosen the other triangle.