On a number of occasions towards the end of a workshop a child will ask how to make a cube. Then I would have to figure out which one; how much time is left and can we do it without diverting the class from what they are doing.

I tell them yes, but not with the folding we are doing. I show them an easier way using the 4-8 folding rather than 3-6 we have been doing. A cube is made, others see it; very quickly half the class is making cubes without distracting the other half.

Here are five cubes made from both the 4-8 grid, right angle triangles and the 3-6 equilateral triangle grid. They have been colored to the creases used for each folding.

We will explore folding cubes with both the 3-6 and 4-8 symmetries of the circle.

*Below left)* fold the circle in half, and in half again; four points on the circumference, five with the point of intersection from the two perpendicular bisectors. Alternate areas have been colored in to show areas of division.

*Below right)* the inscribed square has been creased and two lines of division parallel to the sides have been folded and alternate areas colored. There are five squares, one inscribed and four divisions of smaller squares.

*Above left)* Two circles of the net above are reformed and joined forming the cube. It is the same as joining two open tetrahedra to form an octahedron; (see http://wholemovement.com/how-to-fold-circles. Scroll down to #3.

*Above right)* is the same forming and joining only with more creases showing more areas colored in. This is pretty straightforward to what we understand about divisions of and construction of the square and cube.

*Below)* is another way to make the cube; it takes a little more folding but is richer in transformational and 2-D designing possibilities.

From folding three diameters we fold the 4-frequency diameter grid. http://wholemovement.com/blog/itemlist/date/2011/2?catid=140.

This grid divides the circle into 12 equal parts that show three differently positioned squares.

*Above)* the grid is shown with one of the squares colored to show that part of the grid that lies inside the square net. Two extra creases were made from the grid to show the axial division perpendicular to the sides. Two folded circles have reformed to a cube. The cubic folds are the same for the 3-6 and 4-8 symmetries, different in context making the divisions from the 3-6 gird uniquely different.

*Above)* the flat circle shows two radii that become the diagonal and edge of the square in different formations. Folding under one-quarter of the circle a right angle tetrahedron is formed. Starting with the folded square the same occurs. They are interchangeable to different scales and forming of the cube.

*Above left*) the circle is formed to a right-angle tetrahedron with perpendicular bisecting diagonals.

*Above right)* the tetrahedron edges are pushed in becoming the diagonals of the squares formed by the three edges of the tetrahedron. There is a reciprocal function between edges and diagonals. The right angle tetrahedron is one-quarter of forming a cube and the three squares are one half of a formed cube.

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*Below)* is another design of division from the 3-6 folded symmetry.

*Below left)* the 4-frequency grid with one folded square. The areas have been alternately filled in to make a more interesting proportional surface design. With these folds there are many possibilities for designing the surface.

*Above right) *two circles joined forming a complete cube.

*Below)* the circumference is folded to the back to show only the square. With out the folded circle it would be difficult to come up with this proportional design.

*Above)* all three creased squares colored to show the three square-compound. By adding creases for the three right angle axis to each square. It generates 24 division in the circle.

*Above)* the circle with circumference folded behind showing only one square. Two squares of the same design are reformed and joined. There is the option of putting six squares together into a cube.

In both the 3-6 and 4-8 folded grids the primary points of intersection represents centers and tangent points of a circle matrix. Curved lines are used to redesign straight creases. These show only two of many possibilities.

*Above left)* the 3-6 triangle grid has been colored using the curves that resemble the closest packing of spheres in a traditional arabesque design.

*Above right)* the 4-8 square grid has been designed the same way.

*Above)* The same design can be “centered” differently in how it lies within the circle. This goes for any grid

*Below)* right and left shows the cube formed from each of the above designs.

*Below)* primary folding possibilities of one circle folded to a 4-8 symmetry where the edge of the inscribed square is divided into 8 equal segments. this is different than starting by folding the diagonal into eight equal divisions.

*Above middle right) *is one half a cube.

*Above bottom right)* shows ¾ of a cube formed from this square division.

A “full” cube can be formed from one circle with the 8-frequency division folded on the diameter/diagonal. Out of infinite possible diameters in the circle the square uses two. We can then see root function is the diameter of the circle which goes for the square as well.

*Above)* Two diagonals/diameters divide the circle into four equal parts. Each diameter is folded to 8 equal divisions. The circle has been reformed into a complete cube where the four part colored areas shows a volumetric division in surface design different than what we usually see.

*Above)* two cubes; one from four circles and one from a single circle are combined making a beautifully proportioned cubic system from five circles. Pattern blocks are proportional systems of relationships that have origin in the circle.

The square and cube while they can be constructed separately are relationships of triangles inherent to the circle. There are a lot of possibilities for making your own set of pattern blocks from circles.

*Below) *some three-quarters cubes and some partially formed right-angle tetrahedra make up another kind of proportional cubic system.

Let’s look at other possibilities by folding the circumference to the outside of the cube.

*Below left)* two perpendicular diameters and the square relationship are folded to the 4-8 folded symmetry where the alternate areas in the square are colored.

*Below right)* four circles folded to the right-angle tetrahedron with the circumference folded in, then joined forming a cube.

*Above)* one circle is reformed to the three squares (half cube) with the circumference folded to the outside.

*Above)* the cube from above has been reformed with circumferences on the outside. It reveals the spherical tetrahedron, one of the two tetrahedra forming the cube. The four circles are held together with bobby pins.

*Above)* two views of a different division of proportional folds with circumference on the outside. The cube is smaller with more circumference revealing a spherical truncated tetrahedron.

*Above)* a circle folded to the 3-6 symmetry with alternately areas of one square colored in.

*Above)* Two views of four circles, shown above, have been folded and joined with circumference on the outside. This time part of the second spherical tetrahedron is revealed as they intersect crossing on two opposite faces.

There is another way to form a cube solid is to fold the tetrahedron net; http://wholemovement.com/how-to-fold-circles.

*Below)* the tetrahedron net. Fold the two end points of the three creases that form the inner triangle to the center point and crease (one at a time.) This gives you two more congruent triangles, one on each side of the first folded triangle.

*Above right)* using the creases from either the right or left side of center inscribed triangle, fold the circumference behind forming the triangle. The center triangle will be off center to the just folded triangle. Fold over on second line in from each end point forming a right angle hexagon.

*(below left.)* bring triangle points together forming a right-angle tetrahedron as you would form a regular tetrahedron Tuck triangle flaps inside to hold the right angle tetrahedron together.

*Above right)* the tetrahedron is opened with flaps folded out forming right angles that shows three sides of three-quarters of an open cubic arrangement.

*Above)* two open units joined on surfaces and taped together.

*Above)* two sets of two, above, are joined. The backside cavity is half an octahedron.

*Above)* two sets of four units are joined with the eight open cubic units on the outside. The 24 external points reveal a truncated cube of eight octagon and eight triangle planes. The center is an octahedron axially divided by three intersecting square planes.

There are many variations of cubic division and arrangements, some with circumference folded in, some folded out, and some combined. There is much to be discovered. A cube can be made from one circle or with many, from the 3-6, 4-8, or 5-10 folded circles.

Sometimes students will make a transforming system by hinging four solid cubes together in a square arrangement.

A full cubic transformational system takes eight cubes, all hinged in the same right angle system reflecting the torus ring where the surface rotates through the center opening. You can see an example at:

https://www.facebook.com/photo.php?v=1870781780855&;set=vb.125203137500210&type=3&theater

The surfaces in this torus system were designed to show the closest packing of spheres as well as a few straight-line possibilities. There are other torus systems in the same album made with other forms.

Within the circle are the proportions for blocks having a variety of measures with lot of interesting surface design possibilities.

enjoy...