Construct a Regular Tetrahedron From Congruent Non-regular Tetrahedra

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This Instructables can be regarded as a problem in recreational mathematics, which can be stated as:

construct a regular tetrahedron (a solid with four equilateral triangular faces) from a few (two, three or four) congruent non-rectangular tetrahedra each of which has an equilateral-triangular face.

This problem is related to the problem of constructing a cube (a solid with six square faces; a hexahedron) from six congruent square pyramids (five sided solids with a square base and four triangular faces; a pentahedron). The solution to this problem can be found on the internet. When assembled, all triangular faces of the six pyramids are in contact with a triangular face from an adjacent pyramid, thus leaving no gaps in the constructed cube. The apex of each pyramid lies at the point of intersection of the space diagonals (line segments joining opposite vertices of a cube) of the cube.

This Instructables can also be regarded as an extension of the earlier Instructables as it may 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. For such students, it is useful to construct a transparent tetrahedron showing the positions of its four medians as described in the earlier Instructables. This helps in visualising the dimensions of various non-regular tetrahedra constructed in the present Instructables. The faces of these non-regular tetrahedra have planes passing through one or more medians of a regular tetrahedron.

The first two steps describe:

the method used to obtain triangles that are needed to construct the various non-regular tetraheda;
the nets (defined in Step 2) which can be folded to give the required non-regular tetrahedra.

The remaining three steps show how to divide a regular tetrahedron into:

two non-regular tetrahedra using a single plane passing through one of the tetrahedron’s medians;
three non-regular tetrahedra using a single plane passing through one of the tetrahedron’s medians;
four non-regular tetrahedra using three planes passing through three of the tetrahedron’s medians (this construction is the analogue of the construction which divides a cube into six congruent pyramids).

The only materials needed for this Instuructables are:

thin cardboard (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 rectangles.

It is desirable to have the folowing:

a transparent regular tetrahedron with threads attached showing the positions of its four medians (see the earlier Instructables for constructing this tetrahedron); the description given in this Instructables assumes that such a tetrahedron is available;
although not necessary for illustrating the ideas in this Instructables, one may wish to colour the various non-regular tetrahedra that are made and thus may also require paint or coloured cardstock. (Many of the non-regular tetrahedra shown in this Instructables are coloured using highlighter pens). Also, if available, cardboard packaging which has a single colour other than off-white on one of its faces can be used to make coloured tetrahedra. (One example of this is the use of the black interior surface of a package containing coffee capsules in Step 3 below.)

Step 1: Construction of Triangles Used for Assembling Non-regular Tetrahedra

Description:

All the triangles required in this Instructables are of three types (equilateral, isosceles and right-angled triangles) and are cut out from cardboard rectangles whose dimensions are related to the dimensions of the bases and heights of the triangles. The heights of these triangles can be calculated using Pythagoras’ Theorem (the height of right-angled triangles requires no calculation as their heights are given by the length of one of the sides of the triangles forming the right angle). The procedure used for obtaining all the triangles required in this Instructables is illustrated with the following two examples:

Example 1: Nine equilateral triangles whose side length is s are required to carry out all the steps in this Instructables. Ten such triangles can be obtained by cutting along the solid lines shown in the top rectangle whose length is two and a half times s and whose height is twice (√3)/2 s (the height of an equilateral triangle of side length s is (√3)/2 s. (These lengths were chosen to make use of one of the large faces of a cereal box whose dimensions were approximately 31cm x 22 cm, with s = 10 cm.) As well as obtaining ten equilateral triangles, four right-angled triangles are also obtained whose dimensions are: s, s/2 and (√3)/2 s. Such right angled triangles are required in Step 3. The dashed lines shown in the rectangle are grid lines that help in drawing the solid lines. The dashed vertical lines are spaced at a distance s/2 apart and the one horizontal solid line divides the height of the rectangle into two.

Example 2: Six right angled triangles whose hypotenuse is of length s and whose other two sides are of length s/√3 and √(2/3) s are required in Step 4. They can be obtained by cutting along the lines shown in the bottom rectangle whose length is one and a half times s/√3 and whose height is [√(2/3)]s. The vertical lines are spaced at a distance of s/√3 apart.

All the triangles in this Instructables can be constructed by using the same type of reasoning described in these two examples.

Step 2: Construction of Nets and Non-regular Tetrahedra

Description:

In the context used in this Instructables, a net is a two-dimensional pattern with lines drawn on it that can be folded along these lines to make a model of a three-dimensional solid shape. In this Instructables, nets are constructed by sticking together various triangles using sticky tape to give a two-dimensional pattern. When the net is folded along adjacent sides of triangles joined together by the sticky tape, a three-dimensional model is obtained whose free edges are then joined together with sticky tape.

All the nets used in this Instructables have an equilateral triangle with three other triangles joined by sticky tape to the three sides of the equilateral triangle. Since cereal boxes with labels on one side of the cardboard are used in this Instructables, the photos of the nets shown in the following steps will 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 this is the best stage at which the non-labelled sides of the nets should be painted.

Once the net is constructed and placed on a flat surface, the non-rectangular tetrahedron is made by keeping the equilateral triangle as a base on the flat surface and lifting the other three triangles to form the required non-regular tetrahedron. Sticky tape is then used to hold the non-regular tetrahedron in its correct shape.

Step 3: Constructing Two Congruent Non-regular Tetrahedra

Description:

If available, view a regular tetrahedron with one of its faces in front of you. A plane passing through a median on this face and the edge of the tetrahedron opposite this median divides the tetrahedron into two congruent non-regular tetrahedra. This plane also passes through one of the medians of the regular tetrahedron. The side lengths of the four triangles making up the faces of each of these congruent non-regular tetrahedra are:

an equilateral triangle whose side lengths are equal to s;
two right angled triangles, the length of whose sides are s, s/2 and [(√3)/2] s (the length of the latter side, which is a median of an equilateral triangle of side length s, follows from Pythagoras’ Theorem);
an isosceles triangle whose sides are s, [(√3)/2] s, and [(√3)/2] s.

Use these dimensions to construct two sets of four cardboard triangles as described in Step 1 and arrange each set of four triangles to form two nets and then two non-regular tetrahedra as described in Step 2. One of the above photos shows:

one set of the four triangles;
the four triangles assembled to form a net;
two non-regular tetrahedra viewed with different faces sitting on a flat surface.

These two congruent non-regular tetrahedra can then be assembled to form a regular tetrahedron as follows:

while keeping the equilateral-triangular faces of the two non-regular tetrahedra on a flat surface, move them to a position where one of the edges on each of the two isosceles-triangular faces are adjacent to each other;
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 non-regular tetrahedra upwards until their isosceles-triangular faces are in contact;
holding the two non-regular tetrahedra together in this position, use sticky tape to join together the adjacent edges of the two equilateral-triangular faces that are in contact with each other.

The two isosceles-triangular faces of the non-regular tetrahedra can now be separated so that their equilateral-triangular faces are again sitting on a flat surface. The sticky tape joining the two non-rectangular tetrahedra together now acts as a hinge allowing the model to be opened and closed.

When opened out, the linked equilateral faces of the two non-regular tetrahedra have the shape of a rhombus with opposite angles of 60° and 120° (see above photo). Photos of the constructed regular tetrahedron in two orientations are also shown above.

With the isosceles-triangular face of one of 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 gives one of the heights of the non-regular tetrahedron and can be used along with the area of the isosceles-triangular face to calculate the volume of the non-regular tetrahedron. 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 non-rectangular tetrahedra is s³/(12 √2). This is, as expected, one half the volume of a regular tetrahedron.

Step 4: Constructing Three Congruent Non-regular Tetrahedra

Description:

View the regular tetrahedron with one of its faces sitting on a flat surface. This face is now called the base of the tetrahedron. One of the tetrahedron’s medians passes through the apex of the tetrahedron and the centroid (the point of intersection of the triangle’s three medians) of the equilateral triangle forming the base of the tetrahedron (its length is equal to the height of the tetrahedron). The equilateral triangle forming the base of the tetrahedron can be divided into three congruent isosceles triangles by lines through its three medians. (Do not confuse the median of a tetrahedron with the median of the triangular face of a tetrahedron.) The edges of each of these three isosceles triangles are made up of one side of the equilateral triangle and the lines joining the ends of this side to the centroid of the equilateral triangle. These three isosceles triangles can now be regarded as the bases of three non-regular tetrahedra, which share a common edge at right angles to these bases.

The dimensions of the four triangular faces of the three non-regular tetrahedra are:

an equilateral triangle of side length s;
an isosceles triangle whose sides are s, s/√3, s/√3; the latter two sides are two thirds of the lengths of the medians of the equilateral triangles forming the base of the tetrahedron (the centroid of a triangle divides the medians of a triangle into lengths whose ration is 1:2);
two right angled triangles whose hypotenuses are of length s and whose other sides are of length s/√3 and [√(2/3)] s; the latter side is the length of the height of the regular tetrahedron.

Use these dimensions to construct three sets of four triangles as described in Step 1 and arrange each set of four triangles to form a net and then a tetrahedron as described in Step 2. One of the above photos shows:

one set of the four triangles;
the four triangles assembled to form a net;
two of the three non-regular tetrahedra viewed with different faces sitting on a flat surface.

These three congruent non-regular tetrahedra can then be assembled to form a regular tetrahedron as follows:

while keeping the equilateral-triangular faces of the non-regular tetrahedra on a flat surface, move them so that:

o one of the two edges on the hypotenuses of the right-angled-triangular faces on two of the non-regular tetrahedra touches each of the two edges on the hypotenuses of the two right-angled-triangular faces on the third non-regular tetrahedron;

o the longer sides (those of height [√(2/3)] s of the right-angled-triangular faces adjacent to the right angles meet at a common point;

while keeping these four edges of the right-angled-triangular faces adjacent to each other, lift the two outer non-regular tetrahedra from the flat surface so that the six right-angled-triangular faces of the three non-regular tetrahedra are in contact with each other;
holding the three non-regular tetrahedra together in this position, use sticky tape to join together any two edges of one equilateral-triangular face of a non-regular tetrahedron to a similar edge on the face of two adjacent non-regular tetrahedra.

The three non-regular tetrahedra can now be separated so that their equilateral-triangular faces are again sitting on a flat surface. The two pieces of sticky tape joining the three non-rectangular tetrahedra together now act as hinges allowing the model to be opened and closed.

When opened out, the three linked equilateral-triangular faces of the three non-regular tetrahedra have the shape of a trapezium with three equal sides and the fourth side having a length equal to twice that of the other three sides. Opposite angles of the trapezium are 60° and 120° (see above photo). Photos of the constructed regular tetrahedron in two orientations are also shown above.

With the isosceles-triangular face of one of 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 gives one of the heights of the non-regular tetrahedron and can be used along with the area of the isosceles-triangular face to calculate the volume of the non-regular tetrahedron. Using the equation for the volume of a pyramid given in Step 3, one finds that the volume of each of the non-regular tetrahedra is s³/(18√2) This is, as expected, one third the volume of a regular tetrahedron.

Step 5: Constructing Four Congruent Non-regular Tetrahedra

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By viewing a regular tetrahedron with any one of its faces directly in front of you, you should see that the tetrahedron’s three medians joining the vertices of the face directly in front of you meet at a common point which can be regarded as the apex of a non-regular tetrahedron whose base is the face of the regular tetrahedron facing you. The planes of the three other faces of this non-regular tetrahedron pass through the medians of the regular tetrahedron emanating from the vertices of the tetrahedral face directly in front of you. This non-regular tetrahedron has three congruent faces adjacent to each edge of this base. The dimensions of the triangles making up the faces of this non-regular tetrahedron are:

one equilateral triangle whose side length is s;
three isosceles triangles with two sides of length (s√3)/(2√2) (three quarters the height of the regular tetrahedron; the point of intersection of the four medians of a regular tetrahedron divide each median into lengths whose ratio is 1:3) and the third side of length s.

Use these dimensions to construct four sets of four cardboard triangles as described in Step 1 and arrange each set of four triangles to form a net and then a tetrahedron as described in Step 2. One of the above photos shows:

one set of the four triangles;
the four triangles assembled to form a net;
two of the four non-regular tetrahedra viewed with different faces sitting on a flat surface.

These four congruent non-regular tetrahedra can then be assembled to form a regular tetrahedron as follows:

while keeping the equilateral-triangular faces of the four tetrahedra on a flat surface, move three of them so that any one edge of the equilateral-triangular face of each of the three non-regular tetrahedra is adjacent to one of the three edges on the equilateral-triangular face of the fourth non-regular tetrahedron. This results in three non-regular tetrahedra surrounding a central identical tetrahedron;
while keeping the six adjacent edges of the four non-regular tetrahedra together, lift the three outer non-regular tetrahedra and rotate them so that all the isosceles-triangular faces of the four tetrahedra are in contact;
holding the four non-regular tetrahedra together in this position, use sticky tape to join each of three edges of the central equilateral-triangular face to one of the corresponding edges on each of the three outer equilateral-triangular faces.

The four non-regular tetrahedra can now be separated so that their equilateral-triangular faces are again sitting on a flat surface. The three pieces of sticky tape joining the four non-rectangular tetrahedra together now act as hinges allowing the model to be opened and closed.

When opened out, the equilateral-triangular bases of the four non-regular tetrahedra have an overall shape of an equilateral triangle (see above photo). Photos of the constructed regular tetrahedron in three orientations are also shown above.

Working out the volume of the non-regular tetrahedron for this Step is more complex than in the previous two Steps as it requires knowledge of the height of each non-regular tetrahedron. The height can be calculated from the length of the side of the non-regular tetrahedron’s base (the equilateral triangle) and its edge length (the length of the equal sides of the isosceles triangles) using the equation [(height)² = (edge length)² - (length of side of base)³]. The volume of each of the non-regular tetrahedra is found to be s³/(24√2). Again, as expected, this volume is one fourth the volume of a regular tetrahedron.

Construct a Regular Tetrahedron From Congruent Non-regular Tetrahedra

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