## Introduction: Making a Transparent Regular Tetrahedron for Studying Solid Geometry

Many students studying solid geometry (the geometry of three-dimensional space) have difficulties visualising objects in three dimensions. Jonathan Choate noted in an article in *Geometer’s Corne*r __(http://www.zebragraph.com/Geometers_Corner_files/tetrahedral%20treats.pdf__):

*Solid geometry is often a more complicated subject than plane geometry, for it is undoubtedly more difficult for the mind’s eye to establish and maintain a constant picture of the relevant positions of objects in three dimensional space.*

This *Instructables* presents a simple inexpensive procedure for making a transparent regular tetrahedron (a solid composed of four equilateral triangular faces) using materials readily available in one’s home or that can be purchased cheaply. Such a hands-on model could be useful for students studying solid geometry who have poor innate spatial capabilities and have problems visualising from a two dimensional diagram how an object looks in three dimensions.

While most of us are familiar with the shape and some properties of a cube, we are less familiar with those of the tetrahedron, the simplest Platonic solid (for more information on Platonic solids see https://en.wikipedia.org/wiki/Platonic_solid.

After listing the materials required, this *Instructables* describes a method for constructing a transparent figure (an equilateral triangle) that can be folded into a regular tetrahedron whose edge length is s. Once constructed, the transparent tetrahedron will be used to illustrate the four medians (the lines joining the vertices of a tetrahedron to one of the centers of its opposite faces) and the three bimedians (the lines joining the midpoints of opposite edges of a tetrahedron) of a regular tetrahedron. While two tetrahedra could be made, one illustrating the tetrahedron’s medians and the other its bimedians, the steps described below combine the medians and bimedians in the one tetrahedron. By doing this, the common point of intersection of the medians and bimedians becomes apparent. As an aside, the *Instructables* finishes with an example where the medians of a tetrahedron play a role in chemistry.

## Step 1: Materials Required

- flat pieces of transparent uncoloured flexible plastic (readily obtained from discarded food containers by cutting off any folded edges using scissors);
- thin cardboard (cardboard from cereal boxes was used in this
*Instructables*); - corrugated cardboard for preventing damage of working surfaces when making holes by inserting sharp pointed objects into cardboard and plastic;
- scissors (while a craft or utility knife might give more accurate cuts, for the ideas presented in this
*Instructables*scissors are appropriate); - clear sticky tape;
- removable adhesive tape; (reusable or re-peelable adhesive tape);
- pencil or ballpoint pen;
- a permanent marker pen;
- sewing needle and if needed, a threader;
- two different colours of thread approximately 1mm diameter (crochet thread obtained from a craft shop was used in this
*Instructables*); - a typical school geometry set (containing a ruler, compass, protractor and set square) for drawing triangles;
- an object with a sharp point which allows a hole to be made in the transparent plastic (a pin with a plastic head was used in this
*Instructables*to make an initial hole in the transparent flexible plastic. This hole was enlarged using the pointed arm of a drawing compass).

## Step 2: Construction of Four Transparent Equilateral Triangles

For making a regular tetrahedron whose edge length is *s*, we need four equilateral triangles whose side length is also equal to *s*. For illustrating the geometric properties of the tetrahedron discussed in this *Instructables*, we need to make a hole in the center of each triangle (center here means the point of intersection of the medians of the triangle, a point which is known as the centroid of the triangle). These triangles are constructed as follows:

i. Draw four equilateral triangles of side length *s* on four pieces of soft cardboard (references for drawing equilateral triangles are readily found on the internet);

ii. On each triangle draw lines from the center of each side to the vertex opposite the side (these lines are the medians of the triangles);

iii. Cut out each of the cardboard triangles;

iv. Attach the cardboard triangles (using removable adhesive tape) to pieces of flexible transparent plastic so that the equilateral triangles and their medians drawn on the cardboard can be seen through the plastic (one such example is shown in the above photos);

v. Cut the plastic with scissors, following the sides of the triangles that appear on the cardboard through the plastic sheet;

vi. Using a sharp pointed object, make a hole in each of the triangles at the point of intersection of their three medians (use the medians drawn on the cardboard triangles to indicate where to place the hole);

vii. Using a permanent marking pen place a mark at the center of each edge of each of the triangles.

## Step 3: Construction of a Figure That Folds Into a Transparent Tetrahedron

i. Arrange the four transparent triangles so that one edge of each of three of the triangles is adjacent to one of the edges of the fourth triangle, thus creating a large equilateral triangle; in the following steps we refer to this fourth triangle as the central triangle;

ii. Use clear sticky tape to join each of the three edges of the central triangle to one of the adjacent edges of the three other triangles;

iii. Turn the large triangle over and repeat this latter step placing the sticky tape in the same positions on the reverse face of the large equilateral triangle (this helps to provide some strength to these edges when a hole and thread are inserted through the centers of these edges at a later stage);

iv. Lift up two of the three triangles that are adjacent to the central triangle so that their unattached vertices meet at a point (a vertex of the tetrahedron) and join the two newly created adjacent unattached edges using clear sticky tape.

After performing these steps you should have made a tetrahedron with one of the four faces forming a lid that can be opened and closed via a hinge made of two layers of sticky tape.

## Step 4: Inserting Threads That Will Become the Medians of the Tetrahedron

i. Using one of the colours of the thread, cut four lengths of thread so that each length is approximately equal to twice the length of the medians of the equilateral triangles that were cut out of transparent plastic in Step 2;

ii. Tie a knot at one end of these four pieces of thread;

iii. Using a sewing needle, insert one of the four pieces of thread through one of the holes in the center of one of the faces of the tetrahedron, keeping the knot on the outside surface of the tetrahedron;

iv. Repeat this latter step for each of the remaining three faces of the tetrahedron (including the lid);

v. Except for the thread whose knot lies on the face of the lid of the tetrahedron, perform the following four sub-steps (it may be helpful in performing these sub-steps to open and close the lid of the tetrahedron at various stages during the procedure);

a. Pick up one of the loose pieces of thread, stretch it out so it does not sag, and line it up with the vertex of the tetrahedron opposite the face to which the thread is knotted;

b. At the point where the thread meets the vertex of the tetrahedron, fold it over the vertex and rest the remainder of the thread over an edge where two faces of the tetrahedron meet to form the vertex;

c. Near the vertex, place some clear sticky tape partly over one of the faces of the tetrahedron adjacent to the thread, and with the thread still in place on the edge of tetrahedron, fold the sticky tape so it attaches to the other face of the tetrahedron and holds the thread in place at the vertex;

d. Repeat these three steps for the remaining two vertices of the tetrahedron not opposite the lid of the tetrahedron;

vi. Using a sewing needle and thread, insert the free end of the thread emanating from the lid of the tetrahedron into the vertex opposite its face;

vi. Gently pulling on the free end of the piece of this thread emerging from the vertex, close the lid of the tetrahedron and stretch the thread so it does not sag. Using removable adhesive tape, attach the part of the thread coming out of this vertex as described above in Steps 4 v. b and c.

## Step 5: Observations on the Medians of a Tetrahedron

Each of the four pieces of thread inside the tetrahedron is a median of the tetrahedron. Note that, taking into account all the possible stages where errors of measurement occur in all the previous steps, the four pieces of thread meet at a common point and that the lengths from this common point to a vertex of the tetrahedron and to its opposite face are unequal. Using some more advanced geometrical arguments, it can be shown that the point of intersection of the medians of a regular tetrahedron divides each median in the ratio 1:3, the longer segment being on the side of the vertex of the tetrahedron.

## Step 6: Inserting Threads That Will Become the Bimedians of the Tetrahedron

i. Detach the removable adhesive tape from the vertex of the tetrahedron opposite the lid and open the lid of the tetrahedron making sure you do not pull the thread completely through the vertex;

ii. Attach a piece of removable adhesive tape to the end of the thread that passed through the vertex and attach the end of this thread to a position near the vertex so that in the following steps this thread does not pass through the vertex;

iii. Using thread of a different colour to that used in Step 4, cut three lengths of thread so that each length of thread is one and a half times the length of the edge of the tetrahedron;

iv. Tie a knot at one end of each of the lengths of thread;

v. Using a sewing needle, make holes through the clear sticky tape joining the sides of the equilateral triangles, so that each hole passes through the middle of an edge of the tetrahedron (these holes should approximately correspond with the marks made on the edges of the transparent triangles in Step 2 vii);

vi. Except for the hole in the hinge of the tetrahedron, using a sewing needle insert one of the three pieces of thread through one of the holes made in the sticky tape joins in the previous step (Step 6 v);

vii. Except for the hole in the hinge of the tetrahedron, repeat this latter step (Step 6 vi) for each of the two remaining holes made in the edges of the tetrahedron;

viii. Except for the thread which will pass through the hinge of the tetrahedron, perform the following two sub-steps:

a. Pick up one of the loose pieces of thread, stretch it out so it does not sag, and line it up with the middle of the edge (indicated by the marks drawn on the transparent triangles in Step 2 vii) opposite the edge containing the knot (these two edges are at right angles to each other);

b. Fold the thread over the middle of the edge and with clear sticky tape attach the thread to the face of the tetrahedron on its outside surface;

ix. For the piece of thread which will pass through the hole in the hinge of the tetrahedron, use a needle to insert the thread through this hole; again use clear sticky tape to attach this thread to the surface of the tetrahedron;

x. Detach the removable adhesive tape holding the thread at the vertex of the tetrahedron opposite the lid and close the lid while at the same time gently pulling the thread through the hole in the vertex of the tetrahedron;

xi. Use clear sticky tape to attach this latter piece of thread to the vertex of the tetrahedron as described in Step 4 v. b and c.

## Step 7: Observations on the Bimedians of a Tetrahedron

Each of the three new pieces of thread added to the tetrahedron in Step 6 is a bimedian of the tetrahedron. Again note that these three pieces of thread meet at a common point (within error) and that this time the point of intersection bisects the bimedians of the tetrahedron. Furthermore, the medians and bimedians also meet at a common point.

## Step 8: Final Tidying Up of the Model and Further Observations

The model can be tidied up by using clear sticky tape to join the remaining two unattached edges of the tetrahedron and cutting off any loose pieces of thread passing through the vertices and edges of the tetrahedron.

If one views the model through one of its edges so that this edge is in the same plane as two of the medians (and one of the bimedians) you should be able to rotate the tetrahedron around the opposite edge so that the other two medians are in a plane at right angles to the viewing plane. Therefore a plane passing through two medians emanating from the vertices of an edge of the tetrahedron is at right angles to a similar plane whose medians emerge from the vertices of the edge opposite the first edge.

If one views the model so that two of the bimedians are in the same plane, then the ends of these two bimedians (which pass through the mid points of four edges of the tetrahedron) form a square whose sides are of length *s*/2. The edges of the three squares that are obtained by lining up each pair of bimedians so that they are in the same plane can be shown to form the edges of a regular octahedron (another one of the Platonic solids).

## Step 9: Role of Medians of a Tetrahedron in Chemistry

Understanding the geometry of a regular tetrahedron is of considerable significance (see the Applications section in the *Wikipedia* article https://en.wikipedia.org/wiki/Tetrahedron). For example, in chemistry, the medians of a tetrahedron show the positions of the covalent bonds in molecules like methane with one pair of bonds at right angles to the other pair of bonds. The central carbon atom is at the point of intersection of the four medians.

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