Alan Kay's personal dynamic media concept[1] and its implementation in the interim Dynabook[2], [3] and later the Etoys system propose a mix of written static contents and computer programs. The former is the old world of book, the later the new world of computerised dynamic simulation, model and hand-on experience; put together these contents are the essence of an Active Essay[4]. In this recipe, the Smalltalk language provides the needed flexibility and generality to make this dynamic content actively designed by the user as well.

At a modest scale, Dr. Geo shares this vision of Active Essay, precisely in the field of mathematics and geometry.

## Geometric Active Essay

With mouse operations, the user produces a geometry construction constrained by the inherent properties of the parts defining the construction. A triangle will be designated by three vertices and three segments, then the user instructs Dr. Geo to build the three perpendicular bisectors of each side of the triangle. If Dr. Geo wouldn't know how to do it, the user could have instructed it through a macro-construction. Finally, the user sees the three lines crossing in a common point.

That is more or less the same as the old way of constructing a geometric sketch on a paper-pen environment. From now on, the computerized world of model shows up when the user interacts with his construction; he asks Dr. Geo to drag one of the triangle' vertex and he observes the three lines are always crossing at a same location.

This geometric Active Essay example is only limited by the user imagination and the Dr. Geo construction tools. But what if the user wants more than the provided tools? This is where the Smalltalk dimension of Kay's dynamic content plays its role: provide to the user the needed flexibility to describe its own simulation. For example, what about a dynamic model a teacher designs to show how the Netwon-Raphson algorithm works[5]?

## Programmed Active Essay

To do so, the Dynabook and Dr. Geo can not anticipate the user needs. However enough flexibility is provided by the mean of malleable tools and Smalltalk programming language. From there, in the field of geometry, an Active Essay takes the form of a Smalltalk description to manipulate the Dr. Geo tools. It results on a very short essay, 15 lines, to produce a sketch with a plotted curve and 5 iterations on the Newton-Raphson algorithm.

Active Essay of the Newton-Raphson algorithm

```| sketch f df ptA ptB|
sketch := DrGeoCanvas new axesOn.
f := [ :x | x cos + x ].
df := [ :x | (f value: x + 1e-8) - (f value: x) * 1e8]. "Derivate number"
sketch plot: f from: -20 to: 20.
ptA := (sketch point: 2@0) large; name: 'Drag me'.
5 timesRepeat: [
ptB := sketch point: [ :pt | pt point x @ (f value: pt point x)] parent: ptA.
ptB hide.
(sketch segment: ptA to: ptB) dotted forwardArrow .
ptA := sketch point: [:pt | | x |
x := pt point x.
x - ( (f value: x) / (df value: x) )  @ 0 ] parent: ptB.
ptA hide.
(sketch segment: ptB to: ptA) dotted forwardArrow].
```

More interestingly, it is a model the user can explore with his mouse: zooming in one area, moving the initial value in the algorithm. The learner will visualise the impact of the initial value on the algorithm convergence. It is very tempting to make an analogy with the Archimedes' lever: the Dr.Geo's tools and the Smalltalk lever; with a very short effort, 15 lines, it results on an interactive and dynamic simulation the Dr. Geo's author could never anticipate.

Dr. Geo is developed with Pharo Smalltalk. Dr. Geo itself should be considered as a rather complex Active Essay designed with the tools provided by Pharo and the Smalltalk language. Dr. Geo itself lets the end user access the Pharo tools, making the environment completely circular and transparent. We are very grateful to the talented Pharo community on their continuing effort to improve this fantastic environment. One may not wait long to see the Dynabook and Personal Dynamic Media concepts taking life in Pharo, standing on the shoulder of giants.

Thanks to Chao-Kuei Hung for his editing.

#### Notes

[1] A. Kay, A. Goldberg, Personal Dynamic Media, VPRI Memo, 1977

[2] A. Kay, A Personal Computer for Children of All Ages, Xerox Palo Alto RC, 1972

[3] A. Kay, Afterword: What is a Dynabook?, 2013

[4] T. Yamamiya, A. Warth and T. Kaehler, Active essays on the web, Technical report, 2009

[5] Wikipedia contributors, Newton's method, Wikipedia, The Free Encyclopedia (accessed July 6, 2018)