11/21/14

Events of Addition

The code below was used to construct these events of addition.




The idea is that a gif is used to represent a sequence of events. Ultimately, the act of addition is learned by studying the basic events of addition (addition of single digit numerals), and extending that procedure via a naive procedural and spatial intuition. Students learn to add by observing events of addition and becoming familiar with the form of an additive sequence of events. Later, in more abstract mathematics, a clever student might discover the principles which govern these forms so that they might prove such forms of sequences of events serve the relevant function.

11/20/14

Marks as Assemblies and Assemblies as Marks

In "Assemblies of Marks and Boxes" it was said that a detailed discussion of the foundation of marks and assemblies would be avoided at the moment, but it is for the benefit of utility and clarity that the seemingly "circular" relation between marks and assemblies be addressed in an informal way.

Any mark, when examined closely, may appear as if it is actually an assembly of marks. This is especially true of marks made with electronic displays or modern printers. On a display, or with a printer, there is a "smallest" mark that can be made. On a modern digital screen the smallest mark is called a pixel, and by turning on or off pixels in an assembly of pixels we can create the letters of the alphabet.



It is possible that there is a physical principle which states that there is a smallest mark that can be made, a so-called "quantum mark". A detailed analysis of what such a mark might be leads straight to the forefront of modern physics, and to follow such a line of thought would lead us far astray (at the moment). It is enough, for now, to note that the idea of an indivisible unit of mark is not a foolish approximation, but rather a fitting model of how things appear to be when investigated by experimental science.

As a concrete example of all that has been discussed so far, the following image shows how each of the marks we call capital letters can be represented as assemblies of a single mark (a small green dot).


Thus it is not uncommon, in practice, for an assembly of marks to be identified as a single mark, and not an assembly. Furthermore, and naively, every mark is an assembly of marks containing only itself, and, for those marks we use when writing with pen and paper, it is always possible to recreate each mark as an assembly of tiny dots.

Assemblies of Marks and Boxes

Marks alone are of little interest to anyone. A mark that is given a context and that stands in some naive spatial relation to other marks is capable of changing the world and delighting the mind. Any combination of marks that are near each other on a page form an assembly of marks. Here are some random assemblies of ASCII marks:


There are a few reasons that the general notion of assembly is not used in math and physics. The first is actually more related to marks than assemblies. In the following random assembly there are so many marks that it is impossible to tell which marks do or do not occur within the given assembly.


Since math and physics both deal with clear and exact descriptions of things, the inability to identify clearly and exactly whether a mark does or does not occur in an assembly places a huge restriction on the types of assemblies used in science. Outside of science, the general notion of assembly is used to make, sometimes beautiful, art.


When a mark is on a piece of paper in front of us we can point to it with our finger, and in this way we can pick out a unique mark that belongs to an assembly. In practice, we use bounding boxes to bring our attention to a mark on a page when our finger is not readily available. So for practical purposes, and for the purpose of using marks clearly and exactly, we only use assemblies composed of marks that can be selected uniquely by placing a bounding box around them. Furthermore, we use the same box method when bringing a specific assembly of marks to our attention.

Here is an assembly of visible ASCII marks


Here is an occurrence of each mark in that assembly being selected/box:


The vague idea is this: boxes let us select and organize marks and assemblies on a page. Ultimately it is much easier to use these things than it is to describe how they are used. It is for that reason, and the fact that their use is quite efficient, that we will tentatively avoid a more detailed discussion of the foundations of marks and assemblies.

11/19/14

ASCII Art

I was thinking about moving shapes and whatnot along paths, and suddenly had to make this. It looked a bit slower in my head, but I didn't want to wait long enough for a slower version to compute. The code is below. Each time you run it a new collection of characters, colors, rates, and offsets are chosen. Click for a larger version.

Marks

The most basic tools of math and physics are marks. Sometimes marks are referred to as symbols, glyphs, signs, characters, icons, emblems, tokens, representations, figures, images, ideograms, and much more. In math and physics, many of these words, such as representation and symbol, have a specific meaning that is related to the notion of mark, but which is not exactly the same as mark. For now, a mark is anything that can be made with pen and paper, or any similar set of tools.

In this digital age it is not uncommon for marks to be defined as that which is made upon pressing a key on a computer keyboard. Someone with a more detailed understanding of how computers work might say that a mark is really a collection of pixels on a computer screen. In any of these cases, a mark is that thing that we notice on a page or a screen and which stands out from other things in our field of vision.

This definition is highly informal, but is capable of refinement as we acquire a more detailed knowledge of the other basic tools associated with human studies of math and physics. Rather than dwelling on the logical complexities of defining the notion of mark, we will simply give examples of marks and assume that if you are reading this right now then you are familiar enough with the idea of writing and reading marks to go on to the next part.

In America the first marks that children learn to make in a systematic way are the uppercase and lowercase letters of the alphabet:


When we use marks we usually say that any two marks that look like each other are similar. Similar marks are said to be of the same type. Here are some marks that are similar to one another, that is they are of the same type because they look like each other but they do not look exactly like each other:


Along side the letters of the alphabet, the basic marks taught to children are the decimal digits:


Finally, if we are to collect all of the standard marks that can be made with a modern computer keyboard we must add the following punctuation marks:


Together, all of the marks introduced here form the visible American Standard Code for Information Interchange (ASCII) characters:



It is with these marks that most math and physics is done. There are additional marks that are introduced as needed. The efficient use of marks is of the utmost importance, it is what separates good math and physics from bad math and physics. Ultimately, to get to any meaningful solution in a reasonable amount of time, we must use marks that convey the proper amount of relevant information in the proper way, otherwise we might be left in a sea of nonsensical scribbles.