Wolfram Computation Meets Knowledge

13 Arrays, or Lists of Lists

13Arrays, or Lists of Lists
We’ve seen how Table can be used to make lists. Now let’s see how Table can be used to create higher-dimensional arrays of values.
Make a list of 4 copies of x:
Table[x, 4]
Make a list of 4 copies of a list that contains 5 copies of x:
Table[x, 4, 5]
Use Grid to display the result in a grid:
Grid[Table[x, 4, 5]]
You can use Table with two variables to make a 2D array. The first variable corresponds to the row; the second to the column.
Make an array of colors: red going down, blue going across:
Grid[Table[RGBColor[r, 0, b], {r, 0, 1, .2}, {b, 0, 1, .2}]]
Show every array element as its row number:
Grid[Table[i, {i, 4}, {j, 5}]]
Grid[Table[j, {i, 4}, {j, 5}]]
Generate an array in which each element is the sum of its row and column number:
Grid[Table[i + j, {i, 5}, {j, 5}]]
Generate a multiplication table:
Grid[Table[i*j, {i, 5}, {j, 5}]]
ArrayPlot lets you visualize values in an array. Larger values are shown darker.
Visualize a multiplication table:
ArrayPlot[Table[i*j, {i, 5}, {j, 5}]]
Generate and plot an array of random values:
ArrayPlot[Table[RandomInteger[10], 30, 30]]
ArrayPlot also lets you put colors as values:
ArrayPlot[Table[RandomColor[], 30, 30]]
Images are ultimately arrays of pixels. Color images make each pixel have red, green and blue values. Black-and-white images have pixels with values 0 (black) or 1 (white). You can get the actual pixel values using ImageData.
Find the value of pixels in an image of a “W”:
ImageData[Binarize[ Rasterize["W"] ]]
Use ArrayPlot to visualize the array of values:
ArrayPlot[ImageData[Binarize[ Rasterize["W"] ]]]
The image is of very low resolution, because that’s how Rasterize made it in this case. It’s also white-on-black instead of black-on-white. That’s because in an image 0 is black and 1 is white (like in RGBColor), while ArrayPlot’s default is to make larger values darker.
Find pixel values, then do arithmetic to swap 0 and 1 in the array:
1 - ImageData[Binarize[ Rasterize["W"] ]]
The result is black-on-white:
ArrayPlot[1 - ImageData[Binarize[ Rasterize["W"] ]]]
Table[x,4,5] make a 2D array of values
Grid[array] lay out values from an array in a grid
ArrayPlot[array] visualize the values in an array
ImageData[image] get the array of pixel values from an image
13.1Make a 12×12 multiplication table. »
Expected output:
13.2Make a 5×5 multiplication table for Roman numerals. »
Expected output:
13.3Make a 10×10 grid of random colors. »
Sample expected output:
13.4Make a 10×10 grid of randomly colored random integers between 0 and 10. »
Sample expected output:
13.5Make a grid of all possible strings consisting of pairs of letters of the alphabet (“aa”, “ab”, etc.). »
Expected output:
13.6Visualize {1, 4, 3, 5, 2} with a pie chart, number line, line plot and bar chart. Place these in a 2×2 grid. »
Expected output:
13.7Make an array plot of hue values x*y, where x and y each run from 0 to 1 in steps of 0.05. »
Expected output:
13.8Make an array plot of hue values x/y, where x and y each run from 1 to 50 in steps of 1. »
Expected output:
Expected output:
+13.1Make a 20×20 addition table. »
Expected output:
+13.2Make a 10×10 grid of randomly colored random integers between 0 and 10 that have random size up to 32. »
Sample expected output:
Can the limits of one variable in a table depend on another?
Yes, later ones can depend on earlier ones. Table[x, {i, 4}, {j, i}] makes a “ragged” triangular array.
Can I make tables that are lists of lists of lists?
Yes, you can make tables of any dimension. Image3D gives a way to visualize 3D arrays.
Why does 0 correspond to black, and 1 to white, in images?
0 means zero intensity of light, i.e. black. 1 means maximum intensity, i.e. white.
How do I get the original image back from the output of ImageData?
Just apply the function Image to it.
Next Section