From 17870dac19a29653afd25a08db5d95f90ce334b4 Mon Sep 17 00:00:00 2001 From: Christoph Daniel Schulze Date: Sat, 29 Nov 2014 18:41:05 +0100 Subject: [PATCH] Alles fertig. Alles? 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ztCP74pZbrHE?+o=5g9q-f?473<6Xk6BjMtkReU-I+HZRn9>Pu3|^1aPa>GVC#Rtb6wykuu@ z6Z$M$*#A83QmKD?Ly%=4x+5#ph(l_kYxK6O6)oK`6)H~R{TDK;md@{ajwQG8e_l4L zG*i_H(LQ*Y?6#`RVMKlTc4oIch?P!b5N}dH1YsH-f~(IR&Q5q{7ZG`<7Sz&AzR^z7 q&rh)aOSFmPyGQ617}kW4GqTQ7E%~7d5jmK7{w)>?3JFC?`2Pd>C}sKp diff --git a/main.tex b/main.tex index 13062bc..42f8f5e 100644 --- a/main.tex +++ b/main.tex @@ -114,7 +114,7 @@ Bla. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Kaffeekränzchen und das Kaffeeproblem -\section{Kaffeekränzchen und das Kaffeeproblem} +\section{Erster Hauptteil} \label{sec:kaffeeproblem} Um das Kaffee-Problem betrachten zu können, @@ -336,7 +336,7 @@ analog zu \autoref{def:kaffeekassentransition}. & = & 0. \end{array} \] - Und die bilanzierende Kaffeekasse + Die bilanzierende Kaffeekasse ergibt sich also als \( k = (\Delta_1,\ldots,\Delta_n) \). \end{proof} @@ -378,7 +378,7 @@ analog zu \autoref{def:kaffeekassentransition}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Visualisierung des Kaffeeproblems -\section{Visualisierung des Kaffeeproblems} +\section{Zweiter Hauptteil} \label{sec:visualisierung} Im folgenden stellen wir Möglichkeiten vor, das Kaffeeproblem für Gruppen aus 2 oder 3 Personen @@ -390,7 +390,7 @@ Wie bereits in \autoref{beis:2-kaffee-problem} angesprochen, wird nur eine einzelne Komponente der Kaffeekasse benötigt. Somit stellen wir die Kaffeekasse wie in \autoref{fig:2-kaffee-problem} dar. -\begin{figure}[htbp] +\begin{figure}[tb] \centering \includegraphics[scale=1]{2KaffeeProblem} \caption{% @@ -413,71 +413,79 @@ Analog zu \autoref{ssec:2-kaffee-problem} kann für die Darstellung des 3-Kaffee-Problems ein kartesisches Koordinatensystem verwendet werden, in dem die Werte von \(\Delta_1\) und \(\Delta_2\) -als Koordinaten eingetragen werden. +als Koordinaten eingetragen werden +(\autoref{fig:3-kaffee-kartesisch}). +Der sich ergebende Wert von \(\Delta_3\) +kann auf den dort eingetragenen Diagonalen abgelesen werden, +deren Wert nach links steigt und nach rechts fällt. -\begin{figure}[htbp] +\begin{figure}[tb] \centering - \includegraphics[scale=.6]{3KaffeeKartesisch} + \includegraphics[scale=.8]{3KaffeeKartesisch} \caption{% Visualisierung des \(3\)-Kaffee-Problems mit kartesischen Koordianten. + Die Darstellung entspricht der Kaffeekasse \(k = (2,-1,-1) \). } \label{fig:3-kaffee-kartesisch} \end{figure} -In \autoref{fig:3-kaffee-kartesisch} ist eine -kartesische Darstellung für die Kaffeekasse -\(k = (2,-1,-1) \) gegeben. -Die Aktualisierung der Kaffeekasse, -und das Ablesen der Lösung des 3-Kaffee-Problems, -ist hier allerdings relativ kompliziert. -Gibt zum Beispiel \(p_1\) nun \(p_2\) einen Kaffee -aus, so muss zunächst im Kopf die -neue Kaffeekasse gebildet werden, -und anschliessend die neuen Werte -für \(\Delta_1\) und \(\Delta_2\) -eingetragen werden. -Zusätzlich ist der Wert von \(\Delta_3\) -nur indirekt zu erkennen. -Dieser ergibt sich aus dem negativen Achensabschnittes -der fallenden Diagonalen, auf der der Punkt -zur Zeit liegt. +Die Aktualisierung der Darstellung muss erfolgen, +indem der Punkt den Änderungen in \(\Delta_1\) und \(\Delta_2\) entsprechend +verschoben wird. +Man kommt hier zuweilen nicht drum herum, +nachzudenken. -\begin{figure}[htbp] +Um dieses schwerwiegende Manko zu beheben +bietet sich eine Visualisierung +in einem gradlinigen, nichtorthogonalen Koordinatensystem an, +wie in \autoref{fig:3-kaffee-schief} dargestellt. +Die Achsen schneiden sich hier in einem Winkel von \(60\degree\). +Die Darstellung unterscheidet sich prinzipiell +nicht sonderlich von der Darstellung im orthogonalen Koordinatensystem. +Allerdings ist das Verschieben der Markierung simpler: +um einen ausgegebenen Kaffee einzutragen +muss die Markierung "`mit dem Kaffee"' +von der ausgebenden Person weg +hin zur empfangenen Person verschoben werden. + +\begin{figure}[tb] \centering - \includegraphics[scale=.6]{3KaffeeProblem} + \includegraphics[scale=.8]{3KaffeeProblem} \caption{% - Visualisierung des \(3\)-Kaffee-Problems mit schiefen Koordianten + Visualisierung des \(3\)-Kaffee-Problems mit schiefen Koordinaten. + Die Darstellung entspricht der Kaffeekasse \(k = (2,-1,-1) \). } \label{fig:3-kaffee-schief} \end{figure} -Als alternative Darstellung kann ein -geradliniges, nichtorthgonales Koordiantensystem -gewählt werden. In \autoref{fig:3-kaffee-schief} -schneiden sich die Achsen -mit einem Winkel von 60\degree. -Bei dieser Art des Koordinatensystems -können die selben Koordinaten verwendet werden, -die bereits im kartesischen Koordinatensystem berechnet wurden. -Allerdings können hier die Namen der Personen an den Seiten -plaziert werden, um eine einfacher Transition -zu ermöglichen. -Nun kann hier auch die Markierung wieder -``mit dem Kaffee'', von der ausgebenden Person weg -und auf die empfangende Person zu, verschoben werden. - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Zusammenfassung -\section{Zusammenfassung} +\section{Schluss} \label{sec:zusammenfassung} -Zusammenfassung des Papers. -Mögliche Future Work: -\begin{itemize} - \item Kaffeeproblem für \(n>3\) visualisieren - \item Wir haben noch keinen Beweis dafür, dass die Visualisierung in einem Dreieck unmöglich oder zumindest blöd ist. -\end{itemize} +In dieser durch die Einleitung eingeleiteten, +durch beiden Hauptteile angereicherten +und im Schluss abgeschlossenen Arbeit +haben wir das n-Kaffee-Problem eingeführt, +Probleme an den Haaren herbeigezogen +und schließlich Visualisierungen entwickelt.\footnote{% +Visualisierungen, die sich übrigens prima an Whiteboards machen.} +Mit Hilfe des Kaffeesatzes konnten wir Kaffeekassen +auf kleinere Kaffeekassen reduzieren +und so auch komplexere Konstellationen visualisieren. +Wir haben bilanzierende und explizite Kaffeekassen +voneinander abgegrenzt +und das Kaffee-Paradoxon nicht nur entdeckt, +sondern auch aufgeklärt. +Offen bleibt, +ob man einfache Visualisierungen auch für \(n\)-Kaffeekränzchen +mit \(n > 3\) entwickeln kann. +Des Weiteren hatten wir keine Lust mehr, +weiter nach noch einfacheren Visualisierungen +für \(n < 4\) zu finden. +Wir überlassen es dem geneigten wissenschaftlichen Nachwuchs, +darüber Köpfe zu zerbrechen. \end{document}