# Polychrony as Chinampas
## 4 Chinampas  
Given an activation diagram, the _profit_ is equal to the number of secondary vertices minus the number of primary vertices. The profit measures the maximum number of extra vertices which are activated as a consequence of the topology of the graph. A connected, non-redundant activation diagram is a _chinampa_ if its profit is greater than or equal to zero. An example of a chinampa is illustrated in Figure 8. The simplest chinampa, in the sense that it involves the least number of vertices, is \(pyramid(3,i,t)\), as shown in Figure 4. The name is due to the similarity of the figures with an ancestral Mexican agricultural technique that uses soil to grow crops on a lake. We imagine that chinampas have crops above the soil, and underneath, there are roots. The profit defines a function from the set of chinampas to nonnegative integers. We denote with \(profit(C)\) the profit of a chinampa \(C\). The Topological Description of a Chinampa  
We describe chinampas over a \(path(l)\). For this goal, we will give the decomposition of a chinampa into pyramids. Remember that we defined \(pyramid(l,i,t)\) as an activation diagram \((B,S)\) where the primary vertices \(S\) are consecutive and have the same time. We extend the definition of a pyramid to the activation diagram \((B,S)\), in which \(S\) contains the secondary vertices. Thus, in chinampa \(C\) of Figure 9, we have two pyramids \(pyramid(3,i,t)\) and \(pyramid(3,i+2,t+2)\), where the vertex \((i+2,t+2)\) is a secondary vertex. Note that a pyramid has only one secondary vertex at the top, which we call the _pyramidin_. If \(P_{1}=(B,S_{1})\) and \(P_{2}=(B,S_{2})\) are pyramids in a chinampa, then we say that pyramid \(P_{2}\) is _stacking into_ pyramid \(P_{1}\) if the pyramidion of \(P_{2}\) is a secondary vertex of \(P_{1}\) other than the pyramidion of \(P_{1}\), and \(P_{2}\) is not contained in \(P_{1}\). Remark 1 shows that the activation diagram of a \(path(l)\) is of the form \(pyramid(k,i,t)\), together with the primary vertices at the right or left of \(pyramid(k,i,t)\). We will see that an activation diagram is a sequence of stacked pyramids:  
Figure 8: Example of a chinampa. **Remark 2**. : _Consider two pyramids \(pyramid(l_{1},i,t)\) and \(pyramid(l_{2},i+l_{1},t)\). The pair of primary vertices \((i+l_{1},t)\) and \((i+l_{1}+1,t)\) activates \((i+l_{1}+1,t+1)\), which is not part of any of the two pyramids. Based on Remark 1, the entire diagonal to which the vertex belongs is activated, as well as the vertical line \(\{(i+l_{1}+1,t+k)\}\) with \(k\in\{1,\cdots,l_{1}+1\}\), and so on. Therefore, we end up with \(pyramid(l_{1}+l_{2},i,t)\). This argument works for two pyramids: \(pyramid(l_{1},i,t)\) and \(pyramid(i+l_{1}+k,i,t)\), separated by activated vertices \(\{(i+l_{1}+j,t)\mid 1\leqslant j\leqslant k\}\). Thus, in such cases, instead of considering several small adjacent pyramids, we always consider only the biggest pyramid that includes the small adjacent pyramids._  
**Proposition 1**. : _There exists only one pyramidion with the maximum time in a chinampa. We call this a spike._  
Proof. : Suppose \((i,t)\) and \((j,t)\) with \(i<j\) are pyramidia with the maximum time. Through connectedness, there is a path from \((i,t)\) to \((j,t)\) with only secondary vertices. Each vertex on the path has a time lower than or equal to \(t\). Starting from the vertices with lower times, we use Remark 2 until we reach those vertices with a time \(t\). At each step, we conclude that all vertices above and between those in the path are secondary vertices. Then, those vertices between \((i,t)\) and \((j,t)\) are activated, meaning that \((i,t)\) and \((j,t)\) are not pyramidia, which is a contradiction. **Remark 3**. : _If \((i,t)\) and \((j,t)\) are activated vertices connected by a path of activated vertices with a time lower than or equal to \(t\), then the argument in the proof of Proposition 1 shows that both of them are in a pyramid whose pyramidion has a time greater than \(t\).