Rectangles Everywhere

We all know how essential the cube of chalk is to cue sports, but how many of us have ever considered the cube-ness of a cube of chalk? Although chalk recipes vary from manufacturer to manufacturer, chalk cube geometric proportions are fairly uniform. A chalk cube is not a geometrically perfect cube, but fortunately for us, the papered bottom of almost any chalk cube is a perfect square and can therefore be used as a portable reference tool. The top of an unused cube of chalk is also a perfect square but becomes less reliable with use, so we will focus on the papered bottom.

The Ghost Cube

To those fortunate to have sight in both eyes, have you ever noticed while focusing on a distant object that nearer objects in your view appear doubled and semitransparent? To see this effect, continue to focus on the screen, but lift a finger to a position between the screen and your eyes. You should notice that your finger appears as two semitransparent fingers—one on the left and the other on the right.

If you keep the focal distance of your eyes at screen distance, you can shift your eyes to the left and to the right and look directly through each of your ghost fingers.

We will make use of this ghost image phenomenon in using the chalk cube as an Angle Detective reference tool. If you have sight in only one eye, the chalk cube can still be used but will appear opaque and be slightly less convenient, because the height of the cube obscures portions of the table. However, in the Angle Detective zipped downloadable folder you will find a printable flat Master Square reference tool that is even more useful than the chalk cube. Of course the printed Master Square would only be allowed during practice sessions.

Important: Before using your chalk cube as a reference in a game situation be sure to check that the rules and everyone involved in the game allow its use. My main intention for sharing this alternative use of a chalk cube is to encourage double checking of your work during practice sessions.

There are several ways to use the bottom square of a chalk cube (or the printed Master Square), but first we must “level” it. We hold the chalk so that the square bottom plane is facing upward and parallel to the table’s playing surface.

If the square is not parallel, its sides and corners do not match the proportions of what would be a square lying flat on the table or at ball height, so as a reference it would yield inaccurate measurements.

You can place a square on the table to practice leveling by hovering the chalk’s bottom square to match. The interior square of the Reference Shot Marker printable tool in the Angle Detective download folder is one option for a perfect square to use.

Another way to practice leveling is to stand at the foot or head of the table and align the left, the right and the head or foot cushions with three adjacent sides of the bottom square. If they can be aligned, then the square is leveled.

By moving the chalk cube closer to or further away from our eyes, we effectively alter the size of its bottom square to fit the Master Square of nearly any shot on the table.

From Position 1, if the cue ball and object ball are near one another, the Master Square is small and we must therefore hold the chalk or portable Master Square tool further away from our eyes so that the square appears small enough to match.

If the balls are far from one another, the Master Square is large and we must hold the chalk or tool near our eyes to make the square appear large enough to match.

You can choose either of the two ghost images to use as the reference. You can also choose to close one eye to see only one opaque square. Sometimes the double image can be distracting.

There are two orientations to hold the chalk cube’s bottom square—Square and Diamond.

Square Orientation

The Square orientation is where one of the cube bottom’s sides is facing us.

One way to use the square orientation is for it to represent the Master or ‘Q’ Square. Carefully position the cube’s bottom square so that one of its ghost images fits a shot’s Master Square. This allows us to vividly see the position of the cue ball, object ball, or pocket relative to the square to determine the shot label.

We can also use the square orientation to represent a subsquare. Carefully position the cube bottom so that one of its ghost images fits precisely within a shot rectangle’s first subsquare.

From there, carefully reposition the square so that it now rests adjacent to the first subsquare.

Continue stacking in this fashion to count the number of subsquares of a shot rectangle. For shots under 45° from Position 1, this technique would mean moving the cube bottom closer or further away from our eyes. For shots over 45°, it would mean moving the cube bottom either right or left.

Diamond Orientation

The Diamond orientation is where one of the cube bottom’s corners is pointing at us and its opposite corner is pointing away from us.

One way to use the diamond orientation is to position the cube bottom so that two of its opposite corners rest at the ends of one of the Master Square’s sides. For under 45° shots from Position 1, this would mean that the corner facing us rests at the Master Square’s near/bottom corner, and the corner furthest from us rests at the object ball.

From here we know that the corner facing us is at ‘A’ and that ‘E’ lies directly below the cube bottom’s right or left corner. We can determine the shot label based on the cue ball’s position relative to ‘A’ and ‘E’. We can even slide the chalk to the right or left by half the diamond’s width to find the shot labels from ‘E’ to ‘Q’.

For over 45° shots, it would mean that the left corner rests at the object ball, and the right or left corner rests at the Master Square’s top/far corner.

From here we know that the right or left corner is at ‘U’ and that ‘S’ lies directly to the right or left of the corner facing us. We can slide the chalk downward by half the diamond’s height to find the shot labels from ‘Q’ to ‘S’.

Another way to use the diamond orientation is to position the cube bottom so that one corner rests at one Master Square corner and an adjacent corner rests at the Master Square’s opposite corner. From Position 1, this would mean that the corner furthest from us rests at the object ball, and the right corner rests at ‘Q’.

From here, ‘A’ and ‘U’ lie directly below or to the right of the far corner, and we vividly see the ‘Q’ reference and what would be a 45° ‘Q’ shot line. We can also determine any shot label based on the cue ball’s position relative to ‘A’ and ‘Q’ or ‘Q’ and ‘U’.

We can even use the diamond orientation for subsquare counting. Position the cube bottom so that one corner rests at one corner of the Master Square, and its adjacent corner rests at the end of the first Zig or Zag of the first subsquare.

From this single placement, the first and second Zigs or Zags of the first and second subsquares can be clearly seen on two adjacent sides of the cube bottom. Counting each diamond as two subsquares, we can stack diamonds by sliding the cube to subsequent corners.

The Square and Diamond orientations work just as well with all Position 1 and Position 2 variations.

Other Chalk Shapes

Chalk comes in shapes other than cubes today such as cylinders and hexagons. These shapes can be used as reference tools as well.

The bottom of a cylinder is a perfect circle which fits neatly into a square. This means that cylindrical chalk can be used for subsquare counting.

Also, drawing a tangent line at the circle’s left, right, top or bottom can still help us find the same key reference labels as with the chalk cube. It becomes something of a Master Circle.

A hexagon is a shape that doesn’t fit neatly into a square, but it is still useful. We can find right angles, 30° ‘E+’, and 60° ‘S-’ references in several areas of a hexagon. We can use these references as we hover the leveled hexagonal chalk over the shot setup.