Worksheet

A set of named expressions and a shared view of what they draw, rendered by <notatio-worksheet>. It is the reactive notebook with two additions, and both fall out of the cells rather than being configured:

• a cell that binds a plain number gets a slider — a knob is just a binding you can move;
• a cell that leaves an axis variable free gets drawn — z is the complex plane and colours as a portrait, x is a curve, x and y a surface.

A binding may be complex, in which case it gets a slider per part.

So there is no control syntax and no plot syntax. s := 2 is a slider because it is a number; PolyLog(s, z) is a portrait because z is the only thing left free in it.

A knob and a view

#The polylogarithm's order

Drag the slider and the portrait follows. The binding cell draws nothing — it has no free axis — so it stays off the screen without being told to.

source

<ClientOnly>
  <notatio-worksheet seed='["s := 2", "PolyLog(s, z)"]' />
</ClientOnly>

Several views, one parameter

#Two conventions, side by side

Both cells read the same s, so they move together. This is what replaces a "which function?" control: the choice is which cells you leave visible, and the mark beside each cell toggles it. Two domain colourings occupy the same plane, so only the upper one is drawn — the other would be entirely hidden behind it.

source

<ClientOnly>
  <notatio-worksheet seed='["s := 2", "HurwitzZeta(s, z)", "Zeta(s, z)"]' />
</ClientOnly>

Curves and surfaces

#A complex knob

A binding does not have to be real. A complex one gets a slider per part, so the Lerch transcendent's order can be swept across the plane rather than along a line.

source

<ClientOnly>
  <notatio-worksheet seed='["w := 0.5 + 8i", "LerchPhi(z, w, 1)"]' />
</ClientOnly>

#The free variables pick the projection

One free x is a curve; x and y together are a surface. Everything visible is drawn into one screen rather than a frame each: the curve sits above the surface, because a lower-dimensional thing hidden under a higher-dimensional one is a thing you cannot see.

source

<ClientOnly>
  <notatio-worksheet seed='["a := 2", "Sin(a * x)", "Sin(a * x) * Cos(y)"]' />
</ClientOnly>

A picture is a value

Image is a picture the way a number is a number — something a cell can evaluate to, and therefore something the screen can composite without knowing where it came from. Rasterize makes one out of a graphic; where pixels are available (Node, the CLI) it renders them, and elsewhere it hands back the document itself, which a page can still draw.

#An image layer

An image stacks like any other plane: it is opaque, so it hides whatever sits beneath it, while the curve drawn over it stays visible.

source

<ClientOnly>
  <notatio-worksheet seed='["Image(\"data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHZpZXdCb3g9IjAgMCAxMjAgNjAiPjxyZWN0IHdpZHRoPSIxMjAiIGhlaWdodD0iNjAiIGZpbGw9IiNlMmU4ZjAiLz48Y2lyY2xlIGN4PSI0MCIgY3k9IjMwIiByPSIxOCIgZmlsbD0iIzM0NTFiMiIvPjxjaXJjbGUgY3g9IjgwIiBjeT0iMzAiIHI9IjE4IiBmaWxsPSIjZDk3NzA2Ii8+PC9zdmc+\")", "Sin(3x)"]' />
</ClientOnly>

Framing is a cell too

A setting is written in a namespace of its own — \mathsf{extent}, which in notatio is the island $\mathsf{extent}$ — so it can never collide with an ordinary name the mathematics is using. Wolfram reserves a context for the same reason.

#Setting the view

Because a setting is an ordinary binding, it gets a slider like any other number — so the framing itself is manipulable. Write Auto instead of a value to hand an axis back to the automatic choice.

source

<ClientOnly>
  <notatio-worksheet seed='["$\\mathsf{extent}$ := 1.2", "PolyLog(2, z)"]' />
</ClientOnly>

The view is one model, not per-renderer options. A 2-D graphic is a camera at a standard distance looking straight at a point, so center and extent are a camera and azimuth/elevation are the angles it happens to sit at — zero and zero for anything flat. Domains are the same statement in the form authors think in: xdomain := (-1, 5) is read back as a centre and a distance.

setting	means
center, extent	where the camera looks, and how much it takes in
xdomain, ydomain	the same, written as ranges
azimuth, elevation	its orientation, in degrees
height	pane height in pixels

Notes

• Panes draw the cell, not its value. A cell is plotted with its bindings substituted but not evaluated, so HurwitzZeta(1, z) draws its pole rather than collapsing to ComplexInfinity and vanishing — and a slider moves a literal without changing the compiled shape.
• Every cell has a mark. A drawable one carries a toggle shaped like what it draws, a binding carries a play button that sweeps it, and anything else carries a sign of what kind of row it is. The rest of that column is the drag handle.
• One screen, stacked. Layers are ordered by dimension — points over curves over planes — and an opaque field hides whatever is beneath it, so only the topmost is drawn at all.
• No history, no ordinals. Every cell is defined by its name and recomputed from its dependencies, so cells can be reordered freely. That is the whole difference from <notatio-notebook>, which keeps an In[n]/Out[n] transcript you can refer back to.
• Bindings are scoped to the sheet. Two sheets on a page, both using x, do not see each other — and neither sees the variables the surrounding prose uses.
• screen places the view: auto (default) puts it beside the cells and drops it below on a narrow viewport, side and below pin it.