What is Quality?

Two things differ quantitatively if a common measurement technique can be applied to both in a numerical way. For example, this puppy and this adult dog differ quantitatively in their weights, dimensions, and other bodily measurements. The adult is a larger, differently proportioned puppy:

In comparison, the stages of life of a butterfly are qualitatively different. The caterpillar, chrysalis, and adult butterfly are so unlike each other, that it is easy to assume they are completely different types of life, a common situation for metamorphosing animals:

The caterpillar, chrysalis, and adult stages of life of the monarch butterfly. Dogs don't do this.

Going beyond number

One way to avoid numbers themselves is to look at infinites. This will allow us to go beyond objects, to directly address domains of action.

One type of manufacturing machine-tool is known as a mill. Mills can be characterized by the number of independent types of motion, the number of degrees of freedom, of their moving cutting implements. This three axis mill can move in the x-y plane, and also move the cutting head up and down. It can make an infinite number of different objects: that is, you couldn't count up all the possibilities and arrive at a particular number.

Three-Axis Mill · CC-BY: Frank L · download high quality video

This five axis mill incorporates two new degrees of freedom: an axial rotation and changing pitch of the cutting head. It can also make an infinite number of different objects, but it includes objects that couldn't have been made by a three-axis mill. Note, in particular, the cutting at the end of this video:

Five-Axis Mill · CC-BY: hermancemachineco · download high quality video

Since the bit on the three-axis mill only goes up and down relative to the piece it is machining, it cannot come in from the side to create overhangs. The objects that can be created by 3-axis milling are sometimes called 2½-dimensional, to contrast them to the increased three-dimensional freedom of a five-axis mill. Comparing the possible products makes this clear. Click the arrows to see the different kinds of 3d shapes:

Slide 1
These gears could have been made with a three axis mill (if the back were flat).
Slide 2
Although the shapes are three-dimensional, there are no overhangs. From this vantage point, every location on the gear has a single “height” towards us.
Slide 3
Could a three-axis mill make these? Possibly, but you'd have to flip each piece over to mill the back as well.
Slide 4
A jet engine. Now, since the turbine blades overlap, repositioning the piece in a three-axis mill could not work. With more than one “height”, they require a five-axis mill.
Example Frame

For either mill, we have an infinite realm of possible produced shapes, but one realm is clearly larger than the other. Instead of looking at the final infinite number of created things, it makes more sense to compare the two by looking at the number of independent modes of action in the two machines — independent modes of action which each have an infinite variety. From this standpoint, the dynamic (rather than spatial) dimensions of the five-axis mill are two greater than the other. This is analogous to the difference between flat two-dimensional painting and the use of perspective. In both cases, the resulting painting is produced on a flat surface, but the idea represented gains true volume in space with perspective; this is a new domain of possiblity for the artist.

Compare these two paintings of the Last Supper, one painted in a Bulgarian monastery in the 16th century, and the other painted by Leonardo da Vinci at the close of the 15th century:

click to enlarge

Both paintings are flat, physically, but da Vinci's occupies space.

Similarly, a hypothetical straight line extends infinitely, as does a flat plane. Yet, there is clearly something different about the plane: it's not that it is infinite, lacking a boundary, since that is true of the line as well. The plane has a greater possibility of different, independent actions. When moving along a line, you can go back and forth, while a plane has back-and-forth as well as side-to-side motion at every location. While “infinite” is a good word to distinguish from something that is clearly finite, it isn’t very helpful for looking at processes like these. Instead, dimensions, or types of connectedness or density make more sense.

Today, technology exists to go beyond the possibilities of cutting, drilling, grinding, and other forms of subtractive machining. Known as additive manufacturing, or 3D printing, this technology adds material layer by layer, meaning it is capable of creating objects with internal cavities, parts within parts, and other forms that could not be produced by milling machines. This video shows a simple 3D printer creating a hyperboloid:

How many degrees of freedom would you say a 3D printer has?

(Additional videos: 3-axis, 5-axis, 3d printer.)

For more on dimensions and degrees of freedom, see this video on Riemann’s Habilitation Dissertation:

Next: Cantor: The Uncountable Continuum, part 1