Many of us think our thoughts using a language of some sort---there is usually some voice in our minds. Language in some ways, makes us who we are. Some even argue in the world of cognitive science that language is the foundation of our consciousness. 


An author who has in their minds representations of intelligent concepts should be able to freely express herself through language with free association---digital expressions of these ideas in some cases requires total control of the computer and all of its processes. 


The vision behind the personal computer was that any person could have full command of the functions of their device. I think this vision has come true to some degree, but not fully when it comes to creating graphics, especially mathematical diagrams online. 


Does the common mathematician or professor have the ability to express concepts through web technology? The Web has its own language, and the goal of this project is to help blur the lines between what authoring the mathematical Web should be like and typesetting beautiful Math.


If you know \LaTeX, then get ready to author interactive diagrams in real-time (try using mouse or touch to interact with diagrams).


\begin{interactive}
What matters most is minimizing the distance between our expression and our execution of an idea. For example, I can describe a vector at $(0,0)$ and initial value of the head at $(2,2)$ that will follow a user touch or mouse event. This will produce the following interaction:


\begin{center}
\begin{pspicture}(-2,-2)(2,2)
\psframe(-2,-2)(2,2)
\userline[linewidth=1.5 pt]{->}(0,0)(2,2)
\end{pspicture}
\end{center}


This was as easy as using this \TeX, which many math professors could understand.

\begin{verbatim}
\begin{pspicture}(-2,-2)(2,2)
\psframe(-2,-2)(2,2)
\userline[linewidth=1.5 pt]{->}(0,0)(2,2)
\end{pspicture}
\end{verbatim}

You can start to add more lines, if you wish:


\begin{center}
\begin{pspicture}(-2,-2)(2,2)
\psframe(-2,-2)(2,2)
\userline[linewidth=2pt,linecolor=green]{->}(0,0)(2,2){-x}{-y}
\userline[linewidth=2pt,linecolor=red]{->}(0,0)(2,2){0}{y}
\userline[linewidth=2pt,linecolor=purple]{->}(0,0)(2,2){-x}{cos(y)}
\userline[linewidth=2pt,linecolor=lightblue]{->}(0,0)(2,2)(sin(x)}{-y}
\end{pspicture}
\end{center}


I can also draw a more complex version, and start to make more useful diagrams to describe vectors:

\begin{center}
\begin{pspicture}(-5,-5)(5,5)
        
% y-axis
\rput(0.3,3.75){ $Im$ }
\psline{->}(0,-3.75)(0,3.75)

% x-axis
\rput(3.75,0.3){ $Re$ }
\psline{->}(-3.75,0)(3.75,0)
			
% the circle
\pscircle(0,0){ 3 }
			

 % new vector 
\rput(2.3,1){$e^{i\omega}-\alpha$}
\userline[linewidth=1.5 pt]{->}(1.500,0.000)(2.121,2.121)
\userline[linewidth=1.5 pt,linecolor=blue]{->}(0,0.000)(2.121,2.121){(x>0) ? 3 * cos( atan(-y/x) ) : -3 * cos( atan(-y/x) ) }{ (x>0) ? -3 * sin( atan(-y/x) ) : 3 * sin( atan(-y/x) )}

\userline[linewidth=1.5 pt,linestyle=dashed](-1.500,0.000)(2.121,2.121){x}{0}{x}{y}
\userline[linewidth=1.5 pt,linestyle=dashed](-1.500,0.000)(2.121,2.121){0}{y}{x}{y}

\rput(-0.75,-4.25){$1+\alpha$}
\rput(2.25,-4.25){$1-\alpha$}
\psline{<->}(-3,-4)(1.5,-4)
\psline{<->}(1.5,-4)(3,-4)
\psline[linestyle=dashed](3,-4.5)(3,0)
\psline[linestyle=dashed](-3,-4.5)(-3,0)
\psline[linestyle=dashed](1.5,-4.5)(1.5,0)

	
\end{pspicture}
\end{center}

Think about the student learning a new concept. Here is an example of teaching integration using ``area under the curve'', for example, you can represent $\int_a^b f(x) dx$ as:

\begin{center}
\begin{pspicture}(-4,-3)(4,3)
		
\uservariable{alpha}(0,0){x}

\psplot[algebraic,linewidth=2pt,fillstyle=solid, fillcolor=lightblue]{-4}{alpha}{sin(x)}
\psplot[algebraic,linewidth=2pt]{-4}{4}{sin(x)}
	
\psline{->}(-4,0)(4,0)

    
\end{pspicture}
\end{center}

The next example allows you to move the height of the graph and also integrate over a moving interval.

\begin{center}
\begin{pspicture}(-4,-3)(4,3)

\uservariable{alpha}(0,0){x}
\uservariable{beta}(0,0){y}

\psplot[algebraic,linewidth=2pt,fillstyle=solid, fillcolor=lightblue]{alpha-3}{alpha}{beta + sin(x)}
\psplot[algebraic,linewidth=2pt]{-4}{4}{beta + sin(x)}

\psline{->}(-4,0)(4,0)

\end{pspicture}
\end{center}


\end{interactive}



\begin{print}
What matters most is minimizing the distance between our expression and our execution of an idea. For example, I can describe a vector at $(0,0)$ and initial value of the head at $(2,2)$ that will follow a user touch or mouse event. This will produce the following interaction (you seem to be reading the .pdf, so it will not be interactive! this page is interactive at this url: https://mathapedia.com/books/31/sections/155/405):



\begin{center}
\begin{pspicture}(-2,-2)(2,2)
\psframe(-2,-2)(2,2)
\psline[linewidth=1.5 pt]{->}(0,0)(2,2)
\end{pspicture}
\end{center}


This was as easy as using this \TeX, which many math professors could understand.

\begin{verbatim}
\begin{pspicture}(-2,-2)(2,2)
\psframe(-2,-2)(2,2)
\userline[linewidth=1.5 pt]{->}(0,0)(2,2)
\end{pspicture}
\end{verbatim}

You can start to add more lines, if you wish:


\begin{center}
\begin{pspicture}(-2,-2)(2,2)
\psframe(-2,-2)(2,2)
\psline[linewidth=2pt,linecolor=green]{->}(0,0)(2,2)
\psline[linewidth=2pt,linecolor=red]{->}(0,0)(-2,2)
\psline[linewidth=2pt,linecolor=purple]{->}(0,0)(2,-2)
\psline[linewidth=2pt,linecolor=lightblue]{->}(0,0)(-2,-2)
\end{pspicture}
\end{center}



I can also draw a more complex version, and start to make more useful diagrams to describe vectors:

\begin{center}
\begin{pspicture}(-5,-5)(5,5)
		
% y-axis
\rput(0.3,3.75){ $Im$ }
\psline{->}(0,-3.75)(0,3.75)

% x-axis
\rput(3.75,0.3){ $Re$ }
\psline{->}(-3.75,0)(3.75,0)
			
% the circle
\pscircle(0,0){ 3 }	

 % new vector 
\rput(2.3,1){$e^{i\omega}-\alpha$}
\psline[linewidth=1.5 pt]{->}(1.500,0.000)(2.121,2.121)
\psline[linewidth=1.5 pt,linecolor=blue]{->}(0,0.000)(2.121,2.121)
\psline[linewidth=1.5 pt,linestyle=dashed](-1.500,0.000)(2.121,2.121)
\psline[linewidth=1.5 pt,linestyle=dashed](-1.500,0.000)(2.121,2.121)

\rput(-0.75,-4.25){$1+\alpha$}
\rput(2.25,-4.25){$1-\alpha$}
\psline{<->}(-3,-4)(1.5,-4)
\psline{<->}(1.5,-4)(3,-4)
\psline[linestyle=dashed](3,-4.5)(3,0)
\psline[linestyle=dashed](-3,-4.5)(-3,0)
\psline[linestyle=dashed](1.5,-4.5)(1.5,0)

	
\end{pspicture}
\end{center}

Think about the student learning a new concept. Here is an example of teaching integration using ``area under the curve'', for example, you can represent $\int_a^b f(x) dx$ as:

\begin{center}
\begin{pspicture}(-4,-3)(4,3)

\psplot[algebraic,linewidth=2pt,fillstyle=solid, fillcolor=lightblue]{-3.14}{3.14}{sin(x)}
\psplot[algebraic,linewidth=2pt]{-4}{4}{sin(x)}
	
\psline{->}(-4,0)(4,0)

    
\end{pspicture}
\end{center}

The next example allows you to move the height of the graph and also integrate over a moving interval.

\begin{center}
\begin{pspicture}(-4,-3)(4,3)

\psplot[algebraic,linewidth=2pt,fillstyle=solid, fillcolor=lightblue]{-3.14}{0}{sin(x)}
\psplot[algebraic,linewidth=2pt]{-4}{4}{sin(x)}

\psline{->}(-4,0)(4,0)

\end{pspicture}
\end{center}

\end{print}

Ability to Express Ideas through Technology