Python package simplifies algebraic equations

Beautiful and Colorful

To demonstrate how the two faucets interact while filling the tub, the three bar graphs in Figure 4 represent the respective fill levels by minute. In the graph on the left, the bath user only turns on the slow faucet, which fills the tub in 15 minutes; in the middle, the 10-minute fast faucet is active; and, on the right, both faucets are active together.

Figure 4: Bar charts generated by pyplot for bathtub filling: On the left, the 15-minute tap; in the middle, the 10-minute tap; and on the right, both taps.

In Listing 4, starting in line 5, the fill_tub() function takes the tub fill rate per minute (per_min) as well as the number of values on the X axis displayed (i.e., the minute ticker). Although the scenarios presented each cover different sets of minute values, the matplotlib functions insist that X and Y values be arrays of equal size, or else they will provoke non-intuitive error messages from the depths of the library. The fill_tub() function returns two arrays lx and ly that contain values for the X and Y axes in the bar graph. The two arrays are the same length, and unoccupied Y values with the tub already full are simply set to zero for initialization purposes in line 7.

Listing 4

01 #!/usr/bin/env python3
02 from matplotlib import pyplot as plt
03 from fractions import Fraction
05 def fill_tub(per_min,xmax):
06   lx=list(range(1,xmax+1))
07   ly=[0] * xmax
08   sum=0
09   for i in range(xmax):
10     sum+=per_min
11     if sum > 1:
12       break
13     ly[i]=sum
15   return lx,ly
17 xmax=15
20 fig,ax = plt.subplots(nrows=1,ncols=3,figsize=(15,5))
21 fig.suptitle("Filling a bath tub")
23 lx,ly = fill_tub(Fraction(1,15),xmax)
24 ax[0].bar(lx,ly,color='tab:red')
25 ax[0].set_xlabel("Minutes")
26 ax[0].set_ylabel("Fill Level")
27 ax[0].set_title("15min faucet")
29 lx,ly = fill_tub(Fraction(1,10),xmax)
30 ax[1].bar(lx,ly,color='tab:orange')
31 ax[1].set_xlabel("Minutes")
32 ax[1].set_ylabel("Fill Level")
33 ax[1].set_title("10min faucet")
35 lx,ly = fill_tub(Fraction(1,10)+Fraction(1/15),xmax)
36 ax[2].bar(lx, ly, color='tab:green')
37 ax[2].set_xlabel("Minutes")
38 ax[2].set_ylabel("Fill Level")
39 ax[2].set_title("Both 15min and 10min faucet")
41 plt.savefig("bars.png")

Painting three different graphs side by side in an image file is a piece of cake for matplotlib: The subplots() function simply creates a graph grid with one row as well as three columns in line 20 and returns an array of three graph objects in ax. With these settings, matplotlib manages to arrange the three graphs beautifully, without any manual intervention. Finally, the last line writes the image data to a newly created image file bars.png.

Exact Fraction Calculation

To prevent the program from immediately turning a fraction like 1/15 into a floating-point number, Listing 4 pulls in the fractions package, which handles real fractional arithmetic. Thus, line 23 with Fraction(1, 15) can actually pass 1/15 to fill_tub() rather than, say, 0.066666666667. The latter would cause nasty rounding errors when summing up fractional values later and possibly even cause the tub to overflow. The fractions package, on the other hand, overloads operators like + or >. Therefore, sum += per_min in line 10 actually adds 1/15 to the amount of water in the tub, and sum inside the if condition in line 11 is later exactly 1 when the tub is full, rather than some close, but not quite right floating-point value.

The Author

Mike Schilli works as a software engineer in the San Francisco Bay area, California. Each month in his column, which has been running since 1997, he researches practical applications of various programming languages. If you email him at he will gladly answer any questions.

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