Construct a Regular Tetrahedron From Two Pairs of Congruent Non-regular Tetrahedra Which Are Mirror Images of Each Other

A regular tetrahedron can be regarded as the three-dimensional counterpart of an equilateral triangle. For an equilateral triangle there appears to be only one way to divide the triangle into two, three or four congruent triangles (see photo in Step 1 below). This raises the question:

is there more than one way to divide a regular tetrahedron into two, three or four congruent tetrahedra?

An earlier Instructables showed one way for dividing a regular tetrahedron into two, three and four congruent non-regular tetrahedra with the restriction that each non-regular tetrahedron had one equilateral face. If this latter restriction is removed, then, as shown in this Instructables, there is a second way to divide a regular tetrahedron into four congruent non-regular tetrahedra, thus answering the above question in the affirmative.

This second way of dividing a regular tetrahedron into four congruent non-regular tetrahedra results in having pairs of non-regular tetrahedra that are not identical; although congruent, each of these non-regular tetrahedra is a mirror image of one of the other tetrahedra. Thus a regular tetrahedron can be constructed from two pairs of non-regular tetrahedra with each member of the pair being the mirror image of the other member.

Although not part of the construction described in this Instructables, the first step illustrates the statement made above, namely, that there seems to be only one way to divide an equilateral triangle into two, three or four congruent triangles. This step is then followed with a description of a method used to obtain the dimensions of the various triangles that are required for constructing two pairs of congruent non-regular tetrahedra. This method involves using a transparent regular tetrahedron whose construction was described in an earlier Instructables.The remaining steps show how to:

construct two pairs of nets, each pair of which can be folded into non-regular tetrahedra that are mirror images of each other;
fold the net so as to from two pairs of non-regular tetrahedra, which are mirror images of each other;
assemble the latter four congruent non-regular tetrahedra so as to form a regular tetrahedron.

Rather than repeating descriptions that appear in two previous Instructables, this Instructables assumes that the reader has made a transparent tetrahedron as described in Making a Transparent Regular Tetrahedron for Studying Solid Geometry and understands the procedures used for making cardboard models of non-regular tetrahedra as described in Construct a Regular Tetrahedron From Congruent Non-regular Tetrahedra.

The only materials needed for this Instructables are:

a transparent regular tetrahedron (see the first three steps in the earlier Instructables: Making a Transparent Regular Tetrahedron for Studying Solid Geometry, for constructing this tetrahedron). This tetrahedron has one of its four faces forming a lid that can be opened and closed via a hinge;
thin cardboard (although not necessary, coloured cardstock can be used; cardboard from cereal boxes was used in this Instructables);
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;
pencil or ballpoint pen;
a typical school geometry set (containing a ruler, compass, protractor and set square) for drawing triangles;
although not necessary, coloured paint, if one wishes to colour the various non-regular tetrahedra that one makes; (the non-regular tetrahedra shown in this Instructables are coloured using highlighter pens).

Step 1: Some Properties of an Equilateral Triangle

Although not part of the construction described in this Instructables, this step illustrates the statement made in the introduction that there seems to be only one way to divide an equilateral triangle into two, three or four congruent triangles. The above photo shows an equilateral triangle divided into:

two congruent triangles, using a single median of the triangle (left image in above photo);
three congruent triangles using lines drawn from the intersection of the triangle’s three medians to the triangle’s vertices (middle image in above photo);
four congruent triangles using lines joining the centers of the triangle’s sides (right image in above photo).

Step 2: Working Out the Dimensions of the Four Non-regular Tetrahedra

This step requires making use of a transparent tetrahedron. Carry out the following:

cut out from cardboard two isosceles triangles whose sides have the same dimensions as the isosceles triangles needed to divide a regular tetrahedron into two non-regular tetrahedra (base length equal to s and legs [congruent sides] equal to [(√3)/2]s; see previous Instructables: Construct a Regular Tetrahedron From Congruent Non-regular Tetrahedra);
draw the median on each triangle that passes through the middle of its longest side;
cut a slit in each triangle that starts at a point where the median bisects the longest side and finishes at a point half way along the length of the median (see the leftmost photo above);
insert one triangle into the other one along the cut slits keeping the two triangles at right angles to each other (see second photo from the left above);
insert the two slotted triangles into the transparent tetrahedron so that the longest sides of these triangles lie adjacent to two opposite edges of the tetrahedron; the other sides of the slotted triangles will lie along the medians of each of the four faces of the tetrahedron (see the two photos on the right above).

The transparent (regular) tetrahedron is now divided into four congruent non-regular tetrahedra. The dimensions of the four faces of each of these non-regular tetrahedra can be worked out by examining where the legs of the isosceles triangles and the mid points of their bases touch the faces and edges of the regular tetrahedron. The faces of each of the four congruent non-regular tetrahedra are made up of four triangles whose dimensions are as follows:

two right-angled triangles whose side lengths are [(√3)/2]s (the hypotenuse), s/2 and s/√2 (the distance between opposite edges of a tetrahedron which is also a median of the slotted isosceles triangles and can be found using Pythagoras’ theorem);
two right-angled triangles whose side lengths are s (the hypotenuse), [(√3)/2]s and s/2.

Step 3: Construction of Triangles Used for Assembling Four Non-regular Tetrahedra

Using the dimensions given in Step 2, construct four sets of four cardboard triangles as described in Step 1 of the previous Instructables.

Step 4: Construction of Nets Used for Assembling Four Non-regular Tetrahedra

Unlike the procedure given in the previous Instructables, if one uses cardboard from cereal boxes and wishes not to have the labels on the boxes appearing on the faces of the constructed non-regular tetrahedra, then two different nets are required for constructing two congruent non-regular tetrahedra that are mirror images of each other.

The arrangement of the four right-angled triangles that result in the mirror-image non-regular tetrahedra were found by trial and error; inserting various triangles, constructed in Step 3, into the slotted triangles that were originally inserted into the transparent tetrahedron allowed one to find where the sides of these constructed triangles had to be joined with sticky tape. This resulted in the two nets shown on the left photo above. Each net is made up of one rectangle where the two larger right-angled triangles are joined together and one parallelogram where the two smaller right-angled triangles are joined together. Note that one net is the mirror image of the other net. (One net can be made to coincide with the other net if one member of the pair of nets is flipped over.)

As cereal boxes with labels on one side of the cardboard are used in this Instructables, the above photo of the nets show portions of the labels from the cereal boxes. These labels will not be visible in the constructed non-regular tetrahedra as they will eventually be facing the inside of the tetrahedra when the net is folded into its required three-dimensional shape. If one wishes to have coloured non-regular tetrahedra, then the best stage at which the non-labelled sides of the nets are painted is before the nets are folded into their tetrahedral shapes.

Once the nets are constructed and the nets are folded along adjacent sides of triangles joined together by the sticky tape, three-dimensional models of the non-regular tetrahedra are obtained. Sticky tape is then used to hold these non-regular tetrahedra in their correct shape. The above photos show:

two views of a pair of non-regular tetrahedral which are mirror images of each other with the smaller right-angled triangular faces sitting on a flat surface (see the above middle photo); in this position, the other smaller right-angled triangular face is at right angles to the flat surface;
two views of a pair of non-regular tetrahedral which are mirror images of each other with the larger right-angled triangular faces sitting on a flat surface (see the above photo on the right).

Step 5: Second Way to Construct Four Congruent Non-regular Tetrahedra Part 1

For each pair of non-regular tetrahedra that are mirror images of each other:

place either one of the smaller right-angled triangular faces on each of the non-regular tetrahedra that are mirror images of each other onto a flat surface;
orient the other two smaller right-angled triangular faces on each of the non-regular tetrahedra that are at right angles to the flat surface so that they face you;
line up the shortest side of these latter two right-angled triangular faces so that they are adjacent to each other and have their triangular faces in the one plane; when lined up you should see facing you an isosceles triangle whose legs are of size s (the side length of the regular tetrahedron under construction) and whose height is equal to s/2 (see the above left photo);
with the common side of the two right-angled triangles facing you acting as an axis of rotation, rotate the two non-regular tetrahedra in opposite directions so that the upper two surfaces of the rotated tetrahedra form an equilateral triangle (see the left image in the above photo on the right; the image on the right in this photo shows the same construction for the other pair on non-regular tetrahedra with its isosceles triangular face sitting on a flat surface).

Step 6: Second Way to Construct Four Congruent Non-regular Tetrahedra Part 2

Invert each of the two constructed non-regular tetrahedra so that their equilateral-triangular faces (made up of two right-angled triangles) are sitting on a flat surface. You should now have an isosceles triangle (also made up of two right-angled triangles) as the upper surface of each of the two constructed non-regular tetrahedra. These two constructed congruent non-regular tetrahedra can then be assembled to form a regular tetrahedron as described in the previous Instructables with one modification, which involves the step where sticky tape is used to join these constructed non-regular tetrahedra:

while keeping the equilateral-triangular faces of the two constructed non-regular tetrahedra on a flat surface, move them to a position so that one of the edges on each of the two isosceles-triangular faces are adjacent to each other (see the leftmost photo above);
while keeping these two edges of the isosceles-triangular faces adjacent to each other on the flat surface and using these two edges as an axis of rotation, rotate both the two constructed non-regular tetrahedra upwards until their isosceles-triangular faces are in contact;
holding the four non-regular tetrahedra together in this position, use sticky tape to join together one of the pair of adjacent right-angled triangles that were part of the axis of rotation.

Two images of the constructed regular tetrahedron are shown in the two above photos on the right.

Step 7: Opening and Closing the Regular Tetrahedron

The four non-regular tetrahedra can now be separated in two stages:

separate the two constructed non-regular tetrahedra (each of which is made up of two non-regular tetrahedra that are mirror images of each other) so that their the equilateral-triangular faces are again sitting on a flat surface; the two isosceles-triangular faces of each of these two constructed non-regular tetrahedra are now uppermost;
invert the resulting cardboard solid so that two isosceles-triangular faces of each of these two constructed non-regular tetrahedra (each of which is made up of two non-regular tetrahedra that are mirror images of each other) are now sitting on a flat surface and separate each member of the pair of constructed non-regular tetrahedra that are mirror images of each other (see the left photo above);
line up the resulting four non-regular tetrahedra so that their right-angled triangular faces are all in the same plane as seen in the above photo on the right.

The three pieces of sticky tape joining the four non-rectangular tetrahedra together now act as a hinges allowing the model to be opened and closed.

With any one of the smaller right-angled triangular faces of one of the four the non-regular tetrahedra sitting on a flat surface, there is an edge of this tetrahedron at right angles to the flat surface. The length of this edge (s/2) gives the height of this non-regular tetrahedron when one of its smaller right-angled triangular faces is sitting on a flat surface and this height can be used along with the area of the other smaller right-angled triangular face [s²/(4√2)] to calculate the volume of each of the four non-regular tetrahedra. Using the equation for the volume of a pyramid {Volume = [(area of pyramid’s base) x (height of pyramid from base to apex)]/3} one finds that the volume of each of the four non-rectangular tetrahedra is s³/(24√2). This is, as expected, one quarter the volume of a regular tetrahedron whose edge is of length s.