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using beans to measure volumes

Some of you may greet my question with a scoffing attitude, remembering how callow legislatures made tomato paste a vegetable, such that nutritional guidelines could be met with pizza. An obesity epidemic soon followed, with soaring rates of diabetes, not that pizza was in any way a sole cause.

If that critique resonates, then you may be one of those who suspects “fluff” has been injected, in a corner-cutting way, to remove all the “rigor” from mathematics, which is what the cry babies cry about, if they cry about maths.

Instead of buckling down and learning to concentrate, the fluff lovers try to make it all “fun”. Attention spans suffer. It’s always “on to the next thing”.

Others of you are thinking “of course mathematics is art”. You’re perhaps flashing on intricate Celtic knots and/or Islamic tilings, and seeing right away that mathematics is a goldmine of artifacts marketable as art. Why would anyone question that?

A lot of people don’t realize that handicrafts take skill, time, and concentration to make. Along with problem solving. If the goal is to cultivate a kind of mental stubbornness, a tenacity, a refusal to give up, then one should take up the knitting needles, not lay them down. Or learn to play cards, or focus on board games. None of these folk pass times are about squandering one’s marbles. Sudoku and crossword puzzles serve as exercises to promote mental alacrity. Can’t that be what art is about too?

The skeptics are resonating on another frequency, as their concerns are about pedagogy. If all junior does is watch Youtube and Vimeo movies about the Mandelbrot Set, when will he, she or them learn about the group, ring and field properties of various types with operations?

What about monoids and monads?

What about long division?

The end goal is “delta calculus” (as distinct from “lambda calculus”) and its treatises on limits, convergence versus divergence.

When will junior get all that?

My approach is through the Italian Renaissance and the discovery of perspective drawing. The ability to add a convincing realism around the “missing” Z axis, by means of studies involving parallels, paved the way for CGI (computer generated imagery) in the future.

Prior to the discovery of perspective, 2D canvas artists nevertheless had the tools to suggest landscapes. That which is further away is smaller. That which is in front of something else, relative to the observer, blocks that which is behind it.

We’re able to “read” a painting when it comes to telling foreground apart from background.

Another tool of the Renaissance painter was the grid.

Divide the canvas into squares corresponding to another grid between the painter and the object. The painter would “translate” the colors and shapes in the transparent grid, to the grid on canvas. A completely faithful translation should result in convincing realism.

This hand and brush process later became photography’s notion of photo-realism. A grid of light sensitive materials (like rods and cones in the eye) would pick up color and intensity directly, and could thereby be developed as a faithful rendering.

The photographer only needed to frame (compose) the picture, before making the exposure. The translation was instantaneous, especially when digital technology came into its own. Film, first analog, then digital, followed fast.

That’s all fine and good, when it comes to art history, but do our budding artist mathematicians get the concept of a “vector space” and will they be equipped to follow Maxwell when the fields start emerging? In other words, will their formal understanding of mathematics suffer, for their having been pampered and spoiled with art projects?

The Bourbaki movement in particular marked a rebellion against any reliance on visual aids. Mandelbrot sensed he was in an uphill battle with the mathematics of his day, and if it hadn’t been for computer science and the eminent suitability of his set for computer programming, we might never have gotten much of the fractals literature off the ground.

As some of you know, I favor a two pronged approach when it comes to bridging spatial geometrical thinking with algorithmic procedures.

On a first pass, we avoid coordinate systems completely and simply count the number of balls in specific arrangements. The triangular and square numbers, so-called figurative numbers, mark an entrance to this path. Students get the algebra of N(N+1)/2 as relating to triangles. Then come polyhedral numbers.

Only on a second pass do we get more into grids and coordinate systems, when we’re finally ready to start rendering some of this content on screen.

Midat Gazale’s Gnomon, and The Book of Numbers by Conway and Guy, have been two of the classics our teachers draw from, if you’ve been following our curriculum’s evolution.

The Oregon Curriculum Network has been the headquarters I’ve funded, but when it comes to mom & pop think tanks, I’ve lost track with exactly how many I’ve been able to work with. We’re a thousand dots of light by this time.

Some elements of what I do became more common practice with the spread of Python, the computer language, featuring an elegant implementation of objects. Programming the Mandelbrot Set has become well established as a topic. What better way to introduce the utility of the computer?

Objects come in types as we learn in Category Theory, which have largely replaced or abetted Sets. A set of elements, all of the same type, comes with methods and operations, usually unary and binary.

The operation “to call with” (or “to eat”) is then a basis for lambda calculus. The evaluation of callables may yield new callables. Functions get treated as first class.

However, when introducing objects as coming in types, I’m following Euclid in building up junior’s vocabulary around geometric shapes. The paradigm object is the polyhedron, a spatial shape.

Space provides the initial setting as 2D and 1D simplifications seem more abstract than experiential. We may impose a grid, or tick marks, as a basis for readings, but in all cases the spatial context is still there, as the space of objects.

That’s why a first Python object, beyond a primitive built-in, is very likely a Polyhedron class.

If we’re still doing figurate numbers, then it’s about a stacking. The triangular numbers stack, to give successive tetrahedron numbers.

The sequence 1, 12, 42, 92… reminds us of a growing scaffolding.

We bridge to crystallography for its CCP and HCP packings (also BCC and SCP).

Polyhedrons may be “spun” (rotated), “scaled” (grown and shrunk), and “translated” (moved around). The linear algebra of a vector field kicks in at this point. We’re ready for vectors.

When you scale a tetrahedron by scalar multiplication, multiply its vertex vectors. Volume changes as a 3rd power of linear edge change, area as a second power. We might use a Jupyter Notebook at this stage.

So far, my sequence is not all that radical. Phasing in a computer language is part and parcel of many a contemporary curriculum.

Where my product diverges, perhaps most dramatically, is in adhering to alternative math concepts introduced by the American genius architect, Buckminster Fuller, or at least including them. The geodesic dome is but the tip of the iceberg.

Perhaps owing to the exotic nature of this curriculum, I’ve proposed to explore its relevance and application in some extremely remote circumstances, out of harm’s way, and also thereby rendered harmless, if exposing the general public is of grave concern.

XRL, or eXteme Remote Living (or “livingry” when referring to the equipment), might not be as remote as it sounds, but has to do with creating self sufficiency in many dimensions.

Biosphere 2 is a good example, of a laboratory designed to separate an inside from an outside. The internal atmosphere is like that of a spaceship heading for Mars, not connected to Earth’s. The contained ecosystems have to keep up the gas mix.

A campus in the wilderness, on the site of an old mine, reachable by train over several hours, might count as extremely remote, even without any separation of atmospheres.

The campus at the South Pole likewise counts. You might think of others. I remember Bhutan.

A reason we make this link, between an exotic curriculum and remote sites, is that the Bucky Fuller corpus is all about inventively living with nature and doing more with less. The kinds of research we want to do around domes, for example, requires space to spread out, while city land is at a premium.

Then, have tourists come through to sample the lifestyles and make their own decisions about what to adopt. They’ll see what brands we’re using, to what effect, to manage our ecovillage or campus or whatever.

The facilities won’t be overrun with curious onlookers, given their remoteness. We’ll produce a lot of video.

People will have more of a chance to start over on many levels, in these experimental prototype communities of tomorrow.

By “tourists” I don’t necessarily mean people “on vacation” as these may be “tours of duty” in many ways. Some will be earning academic credit, or credentials of some kind. We might get a badge in a book, like a stamp in your passport.

Some of these remote places might have the look and feel of virtual nations. Or perhaps they seem like refugee camps.

That’s what we call Theater in PATH, and study in Anthropology.

Does any of this XRL stuff mean you won’t be able to follow this curriculum from an office cubicle? Yes and no. Various hands-on activities, such as geocaching, or hiking to the planetarium, will be for campers only.

Not that cities don’t have geocaching.

In truth, a city is just as able to provide maker spaces and testing grounds. The same people will gain experience in many campus digs.

Or think of a theme park. The rides are not simply copies of one another, nor even the simulators, nor do they exactly match the reality we might use them to train for.

A theme park is educational nonetheless, and helps us get ready.

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