You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
author:
- 'Riikka Kangaslampi, Alina Vdovina'
title: 'Triangular hyperbolic buildings.'
---

Introduction {#introduction .unnumbered}
============

A [*polyhedron*]{} is a two-dimensional complex which is obtained from several oriented $p$-gons by identification of corresponding sides. We will consider euclidean and hyperbolic polyhedra. Let’s take a point of the polyhedron and take a sphere (euclidean or hyperbolic) of a small radius at this point. The intersection of the sphere with the polyhedron is a graph, which is called the [*link*]{} at this point.

We consider polyhedra such that all links of all vertices are generalized 4-gons.

We say that a polyhedron $P$ is an $(m,n)$-polyhedron if the girth of any link of $P$ is at least $m$ and each face of $P$ is a polygon with at least $n$ edges.

Let $P$ be a [$(m,n)$-polyhedron such that $m$ and $n$]{} satisfy the inequality $mn \geq 2(m+n)$. This inequality appears, for example, in small cancellation theory [@LS]. The universal covering of a $(m,n)$-polyhedron with the metric introduced in [@BBr p. 165] is a complete metric space of non-positive curvature in the sense of Alexandrov and Busemann [@GH].

In this note we construct examples of $(4,3)$-polyhedra. It follows from [@BS], that the fundamental groups of our polyhedra satisfy the property (T) of Kazhdan. (Another relevant reference is [@Z])

[**Definition.**]{} A [*generalized $m$-gon*]{} is a graph which is a union of subgraphs, called apartments, such that:

-   Every apartment is a cycle composed of $2m$ edges for some fixed $m$.

-   For any two edges there is an apartment containing both of them.

-   If two apartments have a common edge, then there is an isomorphism between them fixing their intersection pointwise.

[**Definition.**]{} Let $\mathcal{P}(p,m)$ be a tessellation of the hyperbolic plane by regular polygons with $p$ sides, with angles $\pi/m$ in each vertex where $m$ is an integer. A [*hyperbolic building*]{} is a polygonal complex $X$, which can be expressed as the union of subcomplexes called apartments such that:

-   Every apartment is isomorphic to $\mathcal{P}(p,m)$.

-   For any two polygons of $X$, there is an apartment containing both of them.

-   For any two apartments $A_1, A_2 \in X$ containing the same polygon, there exists an isomorphism $ A_1 \to A_2$ fixing $A_1 \cap A_2$.

Let $C_p$ be a polyhedron whose faces are $p$-gons and whose links are generalized $m$-gons with $mp>2m+p$. We equip every face of $C_p$ with the hyperbolic metric such that all sides of the polygons are geodesics and all angles are $\pi/m$. Then the universal covering of such polyhedron is a hyperbolic building, see [@Paulin]. The polyhedra we construct in this note are from this class, since $p=3$, $m=4$. Our construction gives examples of hyperbolic triangular buildings with regular triangles as chambers, which were known before, on our knowledge. Examples of hyperbolic buildings with right-angled triangles were constructed by M.Bourdon [@Bourdon].

In the case $p=3$, $m=3$, i.e. $C_p$ is a simplex, we can give a euclidean metric to every face. In this metric all sides of the triangles are geodesics. The universal coverings of these polyhedra with the euclidean metric are euclidean buildings, see [@BB], [@Ba].

Recall (cf. [@Ronan]) that a [*generalized m-gon*]{} is a connected, bipartite graph of diameter $m$ and girth $2m$, in which each vertex lies on at least two edges. A graph is [*bipartite*]{} if its set of vertices can be partitioned into two disjoint subsets such that no two vertices in the same subset lie on a common edge. The vertices of one subset we will call black vertices and the vertices of the other subset the white ones. The [*diameter*]{} is the maximum distance between two vertices and the [*girth*]{} is the length of a shortest circuit.

Let $G$ be a connected bipartite graph on $q+r$ vertices, $q$ black vertices and $r$ white ones. Let $\mathcal{A}$ and $\mathcal{B}$ be two alphabets on $q$ and $r$ letters respectively, $\mathcal{A}=\{x_1, x_2, \ldots, x_q\}$ and $\mathcal{B}=\{y_1, y_2, \ldots,
y_r\}$. We mark every black vertex with an element from $\mathcal{A}$ and every white vertex with an element from $\mathcal{B}$.

Polygonal presentation.
=======================

We recall a definition of polygonal presentation introduced in [@V].

[**Definition.**]{} Suppose we have $n$ disjoint connected bipartite graphs $G_1, G_2, \ldots G_n$. Let $P_i$ and $Q_i$ be the sets of black and white vertices respectively in $G_i$, $i=1,\dots,n$; let $P=\bigcup P_i$, $Q=\bigcup Q_i$, $P_i \cap P_j = \emptyset$, $Q_i \cap Q_j = \emptyset$ for $i \neq j$ and let $\lambda$ be a bijection $\lambda: P\to Q$.

A set $\mathcal{K}$ of $k$-tuples $(x_1,x_2, \ldots, x_k)$, $x_i \in P$, will be called a [*polygonal presentation*]{} over $P$ compatible with $\lambda$ if

-   $(x_1,x_2,x_3, \ldots ,x_k) \in \mathcal{K}$ implies that $(x_2,x_3,\ldots,x_k,x_1) \in \mathcal{K}$;

-   given $x_1,x_2 \in P$, then $(x_1,x_2,x_3, \ldots,x_k) \in \mathcal{K}$ for some $x_3,\ldots,x_k$ if and only if $x_2$ and $\lambda(x_1)$ are incident in some $G_i$;

-   given $x_1,x_2 \in P$, then $(x_1,x_2,x_3, \ldots ,x_k) \in \mathcal{K}$ for at most one $x_3 \in P$.

If there exists such $\mathcal{K}$, we will call $\lambda$ a [*basic bijection*]{}.

Polygonal presentations for $n=1$, $k=3$ were listed in $\cite{Cart}$ with the incidence graph of the finite projective plane of order two or three as the graph $G_1$.

Some polygonal presentations for $k > 3$ were constructed in [@V].

Now we construct polygonal presentations with $k=3$,$n=1$, but the graph $G_1$ is a generalized 4-gon.

$T_1$:

$(x_1,x_2,x_7)$

$(x_1,x_8, x_{11})$

$(x_1,x_{14},x_5)$

$(x_2,x_4,x_{13})$

$(x_{12},x_4,x_2)$

$(x_4,x_9,x_3)$

$(x_6,x_8,x_3)$

$(x_{14},x_6,x_3)$

$(x_{12},x_{10},x_5)$

$(x_{13},x_{15},x_5)$

$(x_{12},x_9,x_6)$

$(x_{11},x_{10},x_7)$

$(x_{14},x_{13},x_7)$

$(x_9,x_{15},x_8)$

$(x_{11},x_{15},x_{10})$

$T_2$:

$(x_1,x_{10},x_1)$

$(x_1,x_{15},x_2)$

$(x_2,x_{11},x_9)$

$(x_2,x_{14},x_3)$

$(x_3,x_7,x_4)$

$(x_3,x_{15},x_{13})$

$(x_4,x_8,x_6)$

$(x_{12},x_{11},x_4)$

$(x_5,x_8,x_5)$

$(x_5,x_{10},x_{12})$

$(x_6,x_{14},x_6)$

$(x_7,x_{12},x_7)$

$(x_{13},x_9,x_8)$

$(x_{14},x_{15},x_9)$

$(x_{13},x_{11},x_{10})$

Let’s show, that these sets are desired polygonal presentations. Remark, that the smallest generalized 4-gon can be presented in the following way: its “points” are pairs $(ij)$, where $i,j=1,...,6$, $i \neq j$ and “lines” are triples of those pairs where all $i,j$ are different. We mark pairs $(ij)$, where $i,j=1,...,6$, $i \neq j$ by numbers from 1 to 15 in natural order. Now one can check by direct examination, that the graph $G_1$ is really the smallest generalized 4-gon. The polygonal presentation $T_1$ and $T_2$ are not equivalent since there is no automorphism of the generalized 4-gon, which transforms one to another.

Construction of polyhedra.
==========================

We can associate a polyhedron $K$ on $n$ vertices with each polygonal presentation $\mathcal{K}$ as follows: for every cyclic $k$-tuple $(x_1,x_2,x_3,\ldots,x_k)$ from the definition we take an oriented $k$-gon on the boundary of which the word $x_1 x_2 x_3\ldots x_k$ is written. To obtain the polyhedron we identify the corresponding sides of our polygons, respecting orientation. We will say that the polyhedron $K$ [*corresponds*]{} to the polygonal presentation $\mathcal{K}$.

[**Lemma [@V]**]{} A polyhedron $K$ which corresponds to a polygonal presentation $\mathcal{K}$ has graphs $G_1, G_2, \ldots, G_n$ as the links.

[**Remark.**]{} Consider a polygonal presentation $\mathcal{K}$. Let $s_i$ be the number of vertices of the graph $G_i$ and $t_i$ be the number of edges of $G_i$, $i=1,\dots,n$. If the polyhedron $K$ corresponds to the polygonal presentation $\mathcal{K}$, then $K$ has $n$ vertices (the number of vertices of $K$ is equal to the number of graphs), $k \sum_{i=1}^n s_i$ edges and $\sum_{i=1}^n t_i$ faces, all faces are polygons with $k$ sides.

For polygonal presentation $T_i,i=1,2$ take 15 oriented regular hyperbolic triangles with angles $\pi/4$, whrite words from the presentation on their boundaries and glue together respecting orientation sides with the same letters. The result is a hyperbolic polyhedron with one vertex and 15 faces and the universal covering is a triangular hyperbolic building. The fundamental group of the polyhedron acts simply transitively on vertices of the building. The group has 15 generators and 15 relations, which come naturally from the polygonal presentation.

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