[MUSIC]. 
In the last lecture, we provided a set of 
three exercises on mapping data types to 
visual attributes. 
In this lecture, we are going to give 
more examples of data combinations and 
the mapping of data to visual attributes. 
And we'll talk about the effects of 
dimensionality on the visual display. 
So, if you have a single variable to 
display or univariate data, you have a 
plethora of choices for visualizations. 
Dot plots, line plots, bar graphs, or 
Tukey box plots are but a few of the 
examples. 
For bivariate quantitative data, the 
scatter plot is the most common choice. 
So, here we have two quantitative 
variables, and the scatter plot describes 
the information quite clearly. 
Moving up to three dimensions, now it's 
getting a bit more challenging. 
So, we can try a 3D scatter plot, but, 
what are some of the problems? 
Let's look at the 3D scatter plot on the 
right with blue cubes. 
Where, exactly, are E and F in relation 
to each other? 
It's a bit of a problem trying to 
represent 3-dimensional information on a 
2-dimensional surface. 
A better choice might be to use a 
2-dimensional scatter plot, with a third 
dimension represented by some other 
attribute. 
Here we use area for the red circles for 
the third quantitative dimension. 
Area's not perceived as accurately as 
position, so you want to put the most 
important dimensions on the x and y axes. 
And save the least important or perhaps 
the most easily distinguished for the 
third. 
Since we already discussed what are the 
issues in trying to map three variables 
to a third spatial dimension. 
Let's look at other ways to deal with 
mapping three variables to a 
visualization. 
So, we know that two variables on the x 
and y axis can map to points. 
So, we have scatter plots, maps, et 
cetera. 
Now the third variable must use something 
like color, size or shape. 
And as you can imagine, we have a very 
large design space. 
So, finding the optimal mapping becomes 
more challenging as the number of 
dimensions increases. 
You can always use shape or different 
types of marks, to represent nominal or 
categorical data. 
Now, moving up to multidimensional data, 
how many variables can be depicted in an 
image? 
We've talked about one, two, and three 
dimensions. 
But what about more? 
Is there some perceptual limit? 
Starting in 1967 Bertin and others asked 
this question. 
And Bertin stated, with up to three rows, 
a data table can be constructed directly 
as a single image. 
However, an image has only three 
dimensions. 
And that barrier is impassible. 
We'll look into answering this question 
in the next lecture.