Dependent clause: "The first power plant built in that country, whose economy at the time was still mostly agrarian, accelerated economic growth in an unprecedented way."  This is perhaps the simplest example of a nonlinear concept in language, in which the dependent clause acts as a grammatical "cul de sac" off of the "through road" of the main clause.

Analogy: "A is to X as B is to Y."  There are two kinds of relationships here:

• The relationship of A to X, analogous to the relationship of B to Y.
• The pair A and B and the pair X and Y, where each pair corresponds to one context.

Multiple, independent adjectives: "Steve-o bought an expensive, pretentious, garish, red suit."  Here, each of the adjectives is independent of the others; they all modify the noun "suit", and therefore the order of adjectives is arbitrary and irrelevant, but are typically governed by language-specific conventions.

• Family tree: "Well, there's me, my mom Kendy, my dad Brudian, my maternal grandma Antopheris, my maternal grandpa Sperrand, my paternal grandma Warshiet, and my paternal grandpa Nart."
• Nesting structures: "On Monday, I'm hanging out with Gionni to take a walk and talk about books, then I've got a dentist appointment; on Tuesday, I'm working all day, then gonna go to Burzber Hall to see The Scinoms and also Hoctone play; on Wednesday, I've got nothing."

Tone of voice -- using a parallel channel (tone) can communicate context.

• "You did such a great job", said earnestly, can be understood to have its obvious, positive meaning, whereas if said with sarcasm, the phrase carries nearly the opposite meaning.  In this case, the significant meaning is carried entirely by flouting a principle of cooperative communication.
• A rising voice tone at the end of a sentence often indicates a question.  Up-talk is another usage of rising voice tone, which is commonly considered to be annoying or dumb-sounding, but could be considered a form of paralanguage which communicates "I will continue speaking".

The essential aspect of a parenthetical/aside is suggested by the word "aside" itself.  The clear way to visually represent this concept is by placing it to the side*, in a separate but parallel body of text, perhaps with an asterisk-type symbol to disambiguate in the case of multiple asides**.

A list of things typically appears as a comma-separated sequence of words with a conjunction (e.g. "and", "or") before the final item in the list.  The conjunction indicates the function of the list (e.g. a sequence of possibilities, a sequence of necessities, etc).

Certain essential elements of the list concept are obscured rendering it as a monolith of text:

• The distinct items in the list are not visually distinct; they are embedded within the paragraph and are only separated by punctuation.
• The order of the items is somewhat obscured by the fact that they are "wrapped" into the lines of the paragraph.
• The "lateness" of the conjuction within the list of items.

Obviously the essential structure of a list is

• the distinct items and
• the order of the items.

A simple visual affordance is to render the list as a sequence of bulletpointed items.  This way, they flow uniformly down the page, each distinct item having its own, clear line.

An argument (logical or rhetorical) consists of a number of statements, each being logically implied by a number of supporting statements, eventually terminating at the foundation of supporting statements, which are taken as assumptions of the argument.  Typically one statement is singled out as the main conclusion, and that is meant as the ultimate product of the argument; the body of supporting statements are just the means to the end.

In conventional writing, the statements of the argument would be rendered in paragraphs of monolithic text, each statement being sequenced no earlier than any of its supporting statements.  Furthermore, each statement must grammatically indicate which of the previous statements are supporting statements, in addition to the essential logic of how the statement is supported. In this monolithic textual form, certain essential features of the argument are obscured:

• Each individual statement is not an obviously distinct visual element, as each manifests as some portion of the paragraph of text, indicated only via punctuation.
• Given a particular statement, the set of supporting statements is not visually indicated, and must be inferred by parsing the statement and potentially even doing more cognitive work to determine which statements are being invoked as support.

However, the actual structure of an argument is that of a directed, acyclic graph (DAG), in which

• the vertices of the graph are statements, and
• there is a directed edge from statement X to statement Y if and only if X is a supporting statement of Y.

Rendering the argument as a DAG makes it visually obvious what the logical progression of statements is, you just follow the arrows.  Visual clarity could be further enhanced if the DAG were arranged such that its arrows were only allowed to point downward (at any angle with a downward component; not horizontal or upward).  From this visual representation, the foundational supporting statements would appear at the top of the visualization, and the ultimate conclusion(s) would appear at the bottom.  Finally, a use of circular logic becomes visually obvious -- it appears as a cycle in the graph, which would require adding an arrow that goes upward -- a big no-no indeed!

In order to utilitize a concept in a sentence in a grammatically correct way, some "padding" has to be used to formally place it within the sentence, when a shorter indicator would not only suffice, but would improve readability, as it would de-emphasize itself and draw attention to the concept being expressed.

Examples:

One more example will be given outside of the table, as it requires more formatting than the table allows.  Example of dialog:

Natural languages could be considered to operate mostly on a subjective and/or metaphorical level, as even the simplest statements are always subject to context and interpretation.  Examples:

• "I'm tired."
• "We all must sacrifice sometimes."
• "I'll see you next Tuesday."

Furthermore, words in natural languages have imprecise and/or multiple definitions. This is totally acceptable to humans because we are contextualizing machines, and can cope with its vagueness in most everyday life situations. The vagueness is probably a good (and maybe even necessary) feature for human use in that it makes a language flexible and doesn't impose an intractable logical consistency constraint. The vagueness is almost certainly the foundation for verbal art forms. But for formal reasoning and quantitative applications, the vagueness of natural languages is a distinct obstacle.  Attempting to carry out formal reasoning using natural languages, especially in dialog with others who may have different context or have different definitional understandings of words, easily leads to unwitting logical non-sequiturs due to the conflation of concepts by limited or imprecise vocabulary.

Making a precise, literal statement in a natural language is difficult.  For example, "I'll always be there for you" has to become some caveat-ridden monstrosity like "If you need help urgently and I'm still alive and I'm not dealing with a more urgent matter of my own definition and I'm physically and mentally able to help you, then I'll help you."  Even then, many of the words are still subject to interpretation and context.

Because of the subjective and metaphorical nature of natural languages, it's not surprising that the reasoning carried out early on using natural languages alone (i.e. in absence of formal or quantitative reasoning) produced anthropocentric, metaphorical views of the nature of reality*, views which are certainly biased by our human cognitive processes.  Arguably, this should be entirely expected of natural languages, as their development was a survival trait, and survival meant, for the majority of human existence, the functioning of a tribe and its attendant social structures.  Accurate, unbiased understanding of the physical world was not of direct use to survival**, as that was not the trait being selected for.

The manifestation of a particular form of structured expression informs which media the form can be put into.  It's possible that a form of structured expression has both static and dynamic manifestations, and potentially multiple of either.

• Static: Manifestation of the form is static and unchanging.  This suitable for any media, but is particularly suitable for physical print.
• Dynamic: Manifestation of the form may change* based on the parameters controlling its presentation.

The interactivity of a particular form of structured expression determines if the form has an interactive interface.  It's possible that a particular structured expression can take both interactive and noninteractive forms, and potentially multiple of either.

• Interactive: Representation or use of the form has an interactive interface.  Some forms may not have a practical noninteractive form, so their interactivity is required.  Examples:
  • A very high resolution image, such as a detailed map of the globe, is far too large to be displayed without some interactive element which allows the user to control the view area and zoom level, as well as what information to display, such as roads, topography, territorial borders, satellite imagery, etc.
  • Some structures are intrinsically high-dimensional, and therefore admit no practical, static representation as a whole, so they require interactive affordances for the user to determine how to slice and view the content.  Examples*.
  • The Level Of Detail structure is best expressed interactively so that viewing and navigation is efficient, instead of requiring the user to skip ahead to a particular page as in a conventional Table Of Contents.
• Noninteractive: The form has a representation without any way to actively modify the presentation.  This is not to say that the form isn't dynamic, for example, a very high resolution image in which one could zoom and scroll, without actually changing the presentation.

The parametrization of a particular form of structured expression is whether or not there are any parameters which govern the form's representation, or if there is essentially only one way to display the form.

• Parametric: The form has a number of parameters which determine how it is rendered*.  Each unique combination of valid parameters gives a distinct representation of that form -- though the content being displayed is not changed.  Most forms of structured expression admit at least some parameters.
• Nonparametric: The form essentially only has one representation that doesn't alter the essential structure.  Examples**.

The availability of tools informs which particular forms of structured expression are practical to use at the current time.  As more tools are developed*, the range of feasibly usable structured expression grows.  Ideally, this category will eventually disappear as tools are developed to cover everything.

• Current: There are well-functioning writing conventions or software tools, currently available to use this particular structured expression.
• Future: The writing conventions or software tools for this particular structured expression are not well-developed or are non-existent, and therefore it's impractical to use this particular structured expression.

Level Of Detail is used in interactive maps and game engines to render visible, spatial objects with a level of detail appropriate to their distance to the viewer.  Think of being able to "zoom in" to take a closer look at the details of the specific things you're interested in.  Examples:

• This article itself presents as an interactive LOD, provided by the collapsing/expanding UI elements.
• An interactive map which allows you to smoothly zoom to any level of detail -- globe, continent, country, province, city, neighborhood, street.
• A Table Of Contents (TOC) of a printed book is a primitive form of non-interactive LOD, in which the tree nodes are presented in depth-first order.  Wikipedia generally presents a hyperlinked TOC for any page that is divided into sections -- this is a primitive form of interactive LOD.
• Computer graphics, especially games, uses LOD in improving rendering efficiency.
• A file manager presents the contents of a computer filesystem via its directory hierarchy, typically offering a view in which directories can be expanded/collapsed to view/hide their contents.
• Computer program source code editor, in which hierarchically organized code blocks can be expanded/collapsed.

The LOD concept generalizes from spatial content to expositional content in that the reader should be able to select the desired level of detail at which they engage the material, "zooming in" to see specific details.  Ideally, this is presented as an interactive affordance.

There are several advantages to the [LOD] tree structure over a conventional, flat/linear structure:

• Hierarchical organization of material -- contrast with a non-hierarchical sequence of sentences or paragraphs.
• Efficient navigation of material -- in the language of complexity theory, one can traverse a balanced LOD tree in $log(n)$ time, where $n$ is the number of nodes in the tree.  Contrast with having to perform a linear search through a sequence of sentences or paragraphs.
• Compact presentation -- the only thing visible is the information that is explicitly requested by use of the expand/collapse UI elements.

Because of the interactive affordance that hides detail unless explicitly asked for, there is less of a need to stay constrained to one point or to a linear reasoning sequence, as would be recommended in a traditional writing convention.  This is a double-edged sword, in that the author can present as many rabbit holes as desired and the reader can choose which ones to go down, but at the same time it provides more opportunities for the exposition to undergo scope creep.

In non-interactive contexts, such as in physical print, a static manifestation of the tree has to be chosen.  A nice set of static representations of trees is given here.  Two notable static representations are:

• Depth-first traversal order -- this corresponds to how a conventional book is organized.  Its Table Of Contents provides a rough tree structure at the beginning of the book, but then the contents of the book are enumerated in order, starting at the beginning of chapter 1, and proceeding at the maximum level of detail.  While printed books are invariably structured using this convention, reading start-to-finish is not an efficient way to gain an understanding of the content.
• Breadth-first traversal order -- this doesn't really appear in conventional writing formats, but is more analogous to an interactive map.  The distinct advantage to this form is that the reader can simply read from the beginning and after completing each level they will have a complete picture of the content to some level of detail.  Analogy*:
  

  Level	Map	Article
  1	Globe	Overall thesis statement
  2	Continent	Rough supporting statements
  3	Country	Sections detailing each statement
  4	City	Subsections with examples, more detail
  5	etc.	etc.

An interesting example of a static LOD is Wittgenstein's Tractatus Logico-Philosophicus, in which each statement is numbered and put into an LOD structure.  Here is a visual representation of its LOD structure in which the levels are given as columns, with nested items aligned in such a way to indicate the tree structure.

Often times there will be more than one channel of exposition presented in parallel.  The word "parallel" implies that there is some correspondence between elements in the parallel channels.  Presenting the channels each as a vertical column in which its content is laid out is a natural and faithful way to visually represent the channels, where corresponding elements are indicated by being placed in vertical alignment with one another.  In an interactive context, the user should be able to hide/show each channel independently.

Examples of parallel channels of exposition:

I think that the mathematical exposition example is one of the more interesting forms of structured expression geared toward technical exposition, as it confers the following qualitative advantages over the conventional academic paper format:

• A formal proof of a mathematical result is provided in what's basically a computer programming language and can be verified by computer with no expert reviewer needed.  This eliminates the need for a referee to manually check the proof.  It also elevates the rigor of the exposition from "acceptable to an expert in the subject"* to objective, mechanically verifiable, and indisputable.
• Given that the formal proof is valid, the "interesting" part of the exposition with respect to publication in a journal is really the narrative it provides.  It contextualizes the result, establishes why it's interesting within the field, describes relevant history, mentions some related open problems, and generally provides the "human" side of the work.  The separation of this from the informal and formal proofs makes it so that a non-expert could read this narrative and understand as much as they possibly could have, without wasting time skipping through the parts of a conventionally-formatted paper that present the proofs or other technical details.
• The informal proof (i.e. the human-written proof in conventional journal article) can concern itself with only the technical details without any extraneous fluff getting in the way.  In my experience, having to intersperse proofs with narrative is annoying and what is considered good exposition depends unstably on the referee's unknown/arbitrary opinions about writing.
• Different mathematicians might be better at the work in the narrative channel, human-written proof channel, or formal proof channel, and the explicit separation between the channels gives a clean delineation between the different types of work.  This gives another kind of specialization which can allow better collaboration between mathematicians, each one focusing on their strength.  Examples**

The idea here is that in most content, there will be many different structures present, describing different aspects of that content.  Each structure will have its own visual representation, and the representation of a particular structure could be thought of as looking at the "projection" of the corresponding aspect of the content.  Examples:

A table is a multiply-indexed data structure (here, "index" and "parameter" are synonyms).  For each valid combination of the specified table parameters, an outcome is specified.  This concept is the direct product of the index sets, which can be thought of as a grid of all combinations of the index values.  In certain situations, not every possible combination will be given.

There are two common ways a table is laid out, but generally it always involves a grid-like visualization.  One visualization directly represents the direct product structure, in which each index set is assigned an axis (vertical or horizontal), and the values are enumerated along each, and their combination forms a grid.  The grid is populated with the content of the table.  Examples of direct product style tables:

• Multiplication table -- the two parameters are the inputs to multiplication, which are presented along the horizontal and vertical axes of the table, and all possible combinations of inputs manifest as the inner grid.
• Matrix notation -- the index sets (the row and column numbers) are usually not displayed, since they're just integers starting at 1.
• Categorization, for example:
  • Partial categorization of forms of structured expression
    

    	Non-interactive	Interactive
    Non-parametric	Bulletpointed list	Level Of Detail article
    Parametric	Sorted set of items in print with fixed sorting criteria	Sorted set of items with GUI to choose one of many sorting criteria

  • Examples of tensors -- the index values are the number of covariant and contravariant tensor components.
• Karnaugh map

If a table has more than 2 indices, then its visualization is a little trickier, and there are different ways to achieve visualization.  One common method is where each element of the table is displayed using an entire row within a grid, and that row explicitly shows the index values as well as the "outcome" value.  Unfortunately, this breaks the visual representation of the direct product structure. Examples of row-based tables:

• Relational database table
• Truth table -- for example, this logical AND truth table

A way to handle tables with more than 2 indices while still attempting to represent the direct product structure visually is to use nested grids of grids.  For example, a truth table for a function of 4 boolean values $f(P,Q,R,S)$ could be rendered as:

The equivalent table rendered as rows would be:

A flow chart presents a visual representation of a decision process or other causal structure. Examples:

• Algorithm
• Decision process
• Understanding flow charts
• Tech tree in a strategy game
• Representation of a novel, history, or a game's story design*
• A choose your own adventure book

Abstract data types (ADTs) could be considered to be the generic building blocks of the structure that structured expression is meant to deal with.  Each ADT has its own essential properties and with those properties natural ways to represent them visually.  Distinct ADTs' natural visualizations may be similar or even the same due to the similarity between the essential properties of those distinct ADTs.  As with most structures, there will typically be many ways to visually represent each ADT.

Presented here are a number of important ADTs.  This is not a comprehensive list by any means.  More examples are given here, though this is also not exhaustive.

There are obvious ways that structured concepts have spatial aspects, such as any numerical or geometric concepts.  However, there are non-obvious ways that structured concepts can naturally translate into a spatial representation.  Here are a few examples of how structured expression manifests spatially:

• Prototypical time series: A sequence can be interpreted as a map/function from ${0, 1, 2, \dots}$ (prototypical time values)* to the numerical values (prototypical measurement values) in order to produce a plot.  There are numerous ways to convey additional information about the function's domain and/or codomain, including:
  • As discrete points -- no information about continuity conveyed.  See also scatter plot.
  • As piecewise-constant segments -- hints that the domain is continuous (e.g. time).  See also bar chart.
  • As a piecewise-linear curve -- hints that both the domain and codomain are continuous, but does not assert any higher order smoothness to the function.  See also line graph.
  • As an interpolated curve -- hints that the sequence is a discrete sampling of a smooth function, but it should be noted that any such interpolation, absent additional information about the nature of the function, only gives an approximation of the values missing from the sequence.
• Prototypical parameterized path -- 2D or 3D: Given 2 or 3 equal-length sequences, then they can be interpreted as coordinate functions for a curve in 2-space or 3-space respectively.  As with the time series, different information about the domain/codomain can be conveyed using different plot types.  Examples:
  • As discrete points (the red points) -- no continuity information.
  • As a piecewise-linear curve -- function is continuous, but no smoothness implied.
  • As an interpolated curve (the blue curve) -- function is visualized using a smooth approximation.
• Translating ordering into direction -- some other structures have some notion of an ordering, which can be encoded visually by mapping it onto a spatial axis.
  • A tree structure -- the order on the tree nodes is given by the successive generations.  Each generation of nodes can be visually aligned to indicate the generational structure.  If the branching factor for the tree is too great, then this visual affordance may not be feasible and perhaps interactivity would be necessary.
  • or a directed acyclic graph (DAG) -- as in the case of the tree structure, there is a notion of generation and therefore of visual alignment to indicate generation, though each node may have multiple ancestors.  Again, as with the tree structure, if the graph is too dense, then an interactive affordance may be needed.
• Distance and difference -- spatial distance between visual elements can be used to indicate distance/difference in the structure.
  • A tree structure can show distance from the root node via sunburst diagram.
  • A graph's connectivity/sparsity can be visualized using a force directed graph.

Again, The Data Visualisation Catalogue is an excellent resource, presenting a categorization of visualization technique based on function.

Instead of a monolithic block of visually indistinct text, the addition of these organizational structures allow for basic, efficient topic-based organization and navigation of written content.  A Table Of Contents could be considered to be a primitive, non-interactive form of Level Of Detail.

The maxim "a picture is worth a thousand words" perhaps doesn't go far enough. Some visualizations really have no faithful rendering into prose, and by convention would fall to using metaphorical language to convey some semblance of the vision to the human reader.  Exceeding the limitation of prose is the crux of the Structured Expression Project Thesis.

Grammatical or semantic content is indicated by the style or the color of text.  Italics is often used to indicate emphasis or use of a foreign word.  Boldface may indicate introduction of a new term.  Use of a different font may indicate a quotation or content belonging to a separate context.  Syntax highlighting is commonly used to make computer program source code more readable, especially regarding comments and numeric/string literals.

Scientific language is a kind of sublanguage which is an invaluable tool, adapting natural languages such that reasonably accurate scientific inquiry and reasoning is possible. This allows natural languages to function on a literal level, away from the vagueness of metaphorical interpretation, and in particular allows for concrete verbal reasoning about an objective reality.

Take for example an exerpt from Newton's Principia (Book I, `Axioms, Or Laws Of Motion`; page 85 in print, which is page 90 of the PDF):

There are no pictures or visual affordances given. This requires the reader to do tremendous cognitive legwork in visualizing the physicality of the description.  Isaac Newton was an intellect nearly unrivaled in all of human history, yet the stilted writing style with its adherence to prose produced an awkward, grammatically incomprehensible mess.  Yet the concept could have been conveyed directly and clearly in a diagram, taking advantage of humans' significant visual and physical reasoning abilities.

Contrast this with the more modern style which intersperses equations, graphs, visualizations, and symbolic affordances. The visualizations deftly take advantage of humans' powerful visual processing faculties, making certain concepts obvious. While symbolic expressions do require the reader to become familiar with the symbolic "vocabulary", they also lend themselves to certain visual regularities that the reader can gradually gain an intuition for quickly understanding them on a cursory level.

These place information which is related to the content in a nearby location in order to avoid interrupting the linear flow of the main written prose. This is perhaps the simplest example of nonlinear structure in the presented content being rendered in a correspondingly nonlinear way -- in other words, not as a dependent clause or parenthetical within the sentence.

The ability to navigate through bodies of writing in a nonlinear manner based on the reader's choice is the essential concept here. References and citations, by which I mean textual references within a book to either another section of the book, or another book entirely, could be considered to be the hand-powered predecessor to hypertext, which necessarily involves an interactive interface, such as a web browser, to navigate to the referent.

The ability to tag content with particular names, indicating an association of that name to the content, thereby facilitating the creation of an association index.  This is an example of a distinct, overlapping structure in addition to whatever structure is already present in the content being tagged.  Examples:

• Stackoverflow content tagged with polymorphism
• Arxiv content tagged for Differential Geometry: math.DG
• The index of a book, in the conventional style of publication