1. Mermaid Diagrams

Folio provides a mermaid environment for text-based diagrams. This is a Folio-specific feature not available in standard LaTeX.

1.1. Flowcharts

\begin{mermaid}
graph TD
    A[Start] --> B{Condition}
    B -->|Yes| C[Process 1]
    B -->|No| D[Process 2]
    C --> E[End]
    D --> E
\end{mermaid}

Result:

graph TD
    A[Start] --> B{Condition}
    B -->|Yes| C[Process 1]
    B -->|No| D[Process 2]
    C --> E[End]
    D --> E

1.2. Horizontal Flowcharts

Use graph LR for left-to-right flow:

\begin{mermaid}
graph LR
    A[Input] --> B[Preprocessing]
    B --> C[Model]
    C --> D[Postprocessing]
    D --> E[Output]
\end{mermaid}

Result:

graph LR
    A[Input] --> B[Preprocessing]
    B --> C[Model]
    C --> D[Postprocessing]
    D --> E[Output]

1.3. Sequence Diagrams

\begin{mermaid}
sequenceDiagram
    participant C as Client
    participant S as Server
    participant D as Database
    C->>S: HTTP Request
    S->>D: Query
    D-->>S: Result
    S-->>C: Response
\end{mermaid}

Result:

sequenceDiagram
    participant C as Client
    participant S as Server
    participant D as Database
    C->>S: HTTP Request
    S->>D: Query
    D-->>S: Result
    S-->>C: Response

1.4. Use Case: Dependency Diagrams

Useful for showing topic dependencies in textbook series:

\begin{mermaid}
graph TD
    A["Sets and Maps"] --> B["Group Definition"]
    B --> C["Subgroups"]
    B --> D["Homomorphisms"]
    C --> E["Cosets"]
    E --> F["Normal Subgroups"]
    F --> G["Quotient Groups"]
    D --> G
    G --> H["Isomorphism Theorems"]

    style A fill:#f5f5f5,stroke:#333,color:#000
    style H fill:#f5f5f5,stroke:#333,color:#000
\end{mermaid}

Result:

graph TD
    A["Sets and Maps"] --> B["Group Definition"]
    B --> C["Subgroups"]
    B --> D["Homomorphisms"]
    C --> E["Cosets"]
    E --> F["Normal Subgroups"]
    F --> G["Quotient Groups"]
    D --> G
    G --> H["Isomorphism Theorems"]

    style A fill:#f5f5f5,stroke:#333,color:#000
    style H fill:#f5f5f5,stroke:#333,color:#000

2. tikzgraph: Graph Drawing

The tikzgraph environment draws graph-theory graphs (vertices and edges). It uses a syntax unified with LaTeX notation for undirected and directed graphs. This is a Folio-specific environment.

2.1. Basic Syntax

Define vertices with id or id/$label$, connect with --, and separate edge chains with ;:

\begin{tikzgraph}
  a/$v_1$ -- b/$v_2$ -- c/$v_3$;
  a -- c;
\end{tikzgraph}

Result:

Omitting the label displays the vertex ID as-is:

\begin{tikzgraph}
  a -- b -- c;
\end{tikzgraph}

Result:

2.2. Directed Graphs

Use -> for directed edges, <- for reverse, and <-> for bidirectional:

\begin{tikzgraph}
  a/$v_1$ -> b/$v_2$ -> c/$v_3$;
  c -> a;
\end{tikzgraph}

Result:

2.3. Edge Styles and Labels

Use square brackets for edge styles (dashed, dotted, thick, double) and labels:

\begin{tikzgraph}
  a/$u$ --[dashed] b/$v$;
  a --[thick, "3"] c/$w$;
  b --[dotted] c;
\end{tikzgraph}

Result:

2.4. Layouts

Choose a layout with the environment option. Default is spring layout (force-directed); circular layout arranges vertices in a circle:

\begin{tikzgraph}[circular layout]
  a/$1$ -- b/$2$ -- c/$3$ -- d/$4$ -- e/$5$ -- a;
\end{tikzgraph}

Result:

2.5. Syntax Summary

2.6. Multi-Edges

Defining multiple edges between the same pair automatically draws them as curves:

\begin{tikzgraph}
  A -- B; A -- B;
  A -- C; A -- C;
  A -- D;
  B -- D;
  C -- D;
\end{tikzgraph}

Result (Bridges of Kö

2.7. Subgraphs

Use {label: ...} to define subgraphs. Nodes within a subgraph are enclosed in a dashed border:

\begin{tikzgraph}
  {SCC $C_1$: a -> b -> c -> a;}
  {SCC $C_2$: d -> e -> d;}
  {SCC $C_3$: f;}
  c -> d;
  e -> f;
\end{tikzgraph}

Result:

Use {a -- b;} without a colon for an unlabeled subgraph.

2.8. Differences from Real TikZ

tikzgraph bears a TikZ-like name but is a Folio-specific simplified graph drawing environment, quite different from real TikZ:

3. tikzcd: Commutative Diagrams

The tikzcd environment draws commutative diagrams, useful for category-theoretic diagrams in math articles.

3.1. Basic Commutative Diagrams

\begin{tikzcd}
A \arrow[r, "f"] \arrow[d, "g"'] & B \arrow[d, "h"] \\
C \arrow[r, "k"'] & D
\end{tikzcd}

Result:

3.2. Arrow Types

\begin{tikzcd}
A \arrow[r, "f"] & B          % Regular morphism
A \arrow[r, hookrightarrow] & B   % Injection
A \arrow[r, twoheadrightarrow] & B % Surjection
A \arrow[r, dashed] & B       % Dashed
\end{tikzcd}

Result:

3.3. Exact Sequences

A short exact sequence:

\begin{tikzcd}
0 \arrow[r] & A \arrow[r, "f"] & B \arrow[r, "g"] & C \arrow[r] & 0
\end{tikzcd}

Result:

3.4. Complex Commutative Diagrams

\begin{tikzcd}
& A \arrow[dl, "p"'] \arrow[dr, "q"] & \\
B \arrow[dr, "f"'] & & C \arrow[dl, "g"] \\
& D &
\end{tikzcd}

Result:

4. Tables (tabular Environment)

4.1. Basic Tables

\begin{tabular}{lcc}
\hline
Name & Symbol & Order \\
\hline
Cyclic group & $\mathbb{Z}/n\mathbb{Z}$ & $n$ \\
Symmetric group & $S_n$ & $n!$ \\
Dihedral group & $D_n$ & $2n$ \\
\hline
\end{tabular}

Result:

4.2. Column Alignment

Use l (left), c (center), r (right):

\begin{tabular}{|l|c|r|}
\hline
Left & Center & Right \\
\hline
apple & banana & cherry \\
dog & elephant & fox \\
\hline
\end{tabular}

Result:

5. figure and table Environments

Use float environments for captions and numbering:

\begin{figure}
\centering
\begin{tikzcd}
A \arrow[r, "f"] \arrow[d, "g"'] & B \arrow[d, "h"] \\
C \arrow[r, "k"'] & D
\end{tikzcd}
\caption{Example of a commutative diagram}
\label{fig:commutative}
\end{figure}

Figure \ref{fig:commutative} shows the basic form of a commutative diagram.

Result:

Figure 1 shows the basic form of a commutative diagram.

\begin{table}
\centering
\begin{tabular}{lcc}
\hline
Group & Commutativity & Finiteness \\
\hline
$(\mathbb{Z}, +)$ & Abelian & Infinite \\
$S_3$ & Non-abelian & Finite \\
$(GL_2(\mathbb{R}), \cdot)$ & Non-abelian & Infinite \\
\hline
\end{tabular}
\caption{Classification of groups}
\label{tab:groups}
\end{table}

Table \ref{tab:groups} shows a basic classification of groups.

Result:

Table 1 shows a basic classification of groups.

6. Images

The \includegraphics command works if you specify an image URL directly:

\includegraphics[width=0.8\textwidth]{https://example.com/image.png}

7. Summary

On Folio, you can draw text-based diagrams with mermaid (flowcharts, etc.), tikzgraph (graph-theory graphs), and tikzcd (commutative diagrams). The tabular environment for tables works as expected. Next, we'll cover cross-references, links, and custom commands.