Modeling Plant FormL-SystemsThe Turtle interpretation of stringsLet’s put our turtle to workNow let’s make it a little bit more complexFASS curves generated from edge-rewriting L-systemsPowerPoint PresentationSlide 83-DAxial TreesSlide 11Let’s build a treeBracketed systemExamples of bracketed systemStochastic L-SystemsContext-sensitive L-SystemsParametric L-SystemsNow for the real stuff…Let’s try to simulate herbaceous plantsA glimpse at the modelsThree Main Type of ModelsPartial L-SystemExamples of cool things in L-system SchemataExamples of cool things in L-System SchemataSlide 24COMPLETE MODELS…MUAHAHAI like flowers!I still like flowers!I hope you aren’t allergic to pollenBut I want more!Other modelsBy changing values for the number of attraction points, the kill distance, influence distance, and the distribution of attraction points…Slide 32Resource Acquisition ModelWait…what’s a binary treeTheir ResultsConclusionReferencesModeling Plant Form Is plant form an emergent property of simple module systems?L-SystemsL-systems are basically a way to rewrite something following a set of rulesFor instance: you have two letters a and b. The rules for rewriting are a->ab and b->aIf we start with a b and start rewriting we get:The Turtle interpretation of stringsSo we have a turtle with a string on its back, the turtle’s state is a triplet (x,y,α). This represents the turtle’s Cartesian coordinates and the angle (α) at which it is traveling.Now, d = step size and ƒ =angle incrementSo we can tell the turtle where to go if we give it directions. We will use the following symbols:F = Move forward by one step length d+ = Turn counterclockwise by angle ƒ- = Turn clockwise by angle ƒLet’s put our turtle to workGiven the axiom w = F-F-F-F and the production successor p = F->F-F+FF-F-F+We can rewrite the phrase n times and tell out turtle to walk.Now let’s make it a little bit more complexEdge rewriting productions substitute figures for polygon edgesFl and Fr represent the turtle obeying the “move forward” command, but now Fl and Fr edges by lines forming left or right turns.These curves can be space-filling and self avoiding (FASS).FASS curves generated from edge-rewriting L-systemsNode rewriting substitutes polygons for nodes on the curveNow we need more things: Entry and exit points (Pa and Qa) and an entry vector and an exit vector (pa and qa)You can also consider an array of m x m square tiles.Each m x m contains a small box inside of it called a frame. Each frame bounds an open self-avoiding polygon.Now when we connect many tiles we will get a macrotile3-DAxial TreesAll of the previous examples were all a single line, but trees are not!An axial tree starts from a base nodeAt each of its nodes there is at most one outgoing straight segmentAll other edges are lateral segmentsA terminal segment is an apexAn axis must:The first segment in the sequence originates from the base or a lateral segment at a nodeEach subsequent segment is straightThe last segment is not followed by any straight segmentSo each axis is a mini axial tree!An axis with all of its descendants is a branchAxes and branchesare ordered as order0 If they originatedAt the base and youCan guess the restLet’s build a treeWe need to have a rewriting mechanism that acts on axial treesOur rewriting rule, or tree production, must replace an edge with an axial treeBracketed systemExamples of bracketed systemNote: The system for addingLeaves to this bush isBiologically whackStochastic L-SystemsSince all plants don’t look the same we will add in some randomization.Context-sensitive L-SystemsWe can make an L-System that show signal propagation so we can send signals from the leaves down or from the roots up. RemovingP2 makesPermanentsignalPlantsReallyUseSignals!Parametric L-SystemsWill help us show time, angles, and irrational line lengths (if d = 1, you cannot express sqrt(2).Is easier than trying to add stuff to non-parametric model.Now for the real stuff…Let’s try to simulate herbaceous plantsEmphasis on space-time relation between plant partsSo there can be flowers and buds on the tree at the same timeInherent capability of growth simulationOur model is good for growing and we can simulate plants at different times and watch how they growLet’s only do herbaceous plants because:The model assumes that the plant controls its own development (endogenous interaction).Herbaceous plants have a lot of directions from their parents (lineage interaction).Woody plants are much more sensitive to their environment, competition among branches and trees, and accidents (exogenous interaction).A glimpse at the modelshttp://algorithmicbotany.org/vmm-deluxe/QT/Greenash/apexview.qthttp://algorithmicbotany.org/vmm-deluxe/QT/Bluebell/field.qtWe can use confocal microscopes to get a real idea of how plants develop and then write a computer model that fits the behaviorWe can also use empirical data on plant development Other models try to use known mechanisms to explain the emergence of plant formsThree Main Type of ModelsPartial L-Systems: Your basic model that is supposed to show us the possible structures of plantsL-System Schemata: Topology and temporal aspects of plants expressed, could help us understand mechanismsComplete L-Systems: Geometric aspects added in (growth rates of internodes, values f branching angles, appearance of organs)Partial L-SystemExamples of cool things in L-system SchemataExamples of cool things in L-System SchemataExamples of cool things in L-System SchemataThis says that the apex (a) produces internodes (I) and leaves (L) [p2]. The time in between growth is m [p1].After delay (d) a signal (s) [p3 an p4]. The signal is sent down the main axis with delay (u) steps per internode (I) [p5 and p7].[p6] removes the signal from the node by using an empty string (e)When the signal reaches the apex (a), the a is transformed into a flowering state (A), which turns into a flower (K) [p8 and p9].Note: u<m or the signal is slower than growth!Plants actually use signals and feedback loops a lot(WUS acts on SAM)!COMPLETE MODELS…MUAHAHAThese are good enough to make imagesWe can tell the model when to make branches using subapical growthPlants actually grow like this!I like flowers!There are a few