How To Draw The Cantor Set In Google Sheets

The Cantor set is a special set of numbers lying between 0 and 1, with some fascinating properties.

It’s created by removing the middle third of a line segment and repeating ad infinitum with the remaining segments, as shown in this gif of the first 7 iterations:

Cantor Set Iterations

The formulas used to create the data for the Cantor set in Google Sheets are interesting, so it’s worth exploring for that reason alone, even if you’re not interested in the underlying mathematical concepts.

But let’s begin by understanding the set in more detail…

What Is The Cantor Set?

Cantor Set in Google Sheets

The Cantor set was discovered in 1874 by Henry John Stephen Smith and subsequently named after German mathematician Georg Cantor.

The construction shown in this post is called the Cantor ternary set, built by removing the middle third of a line segment and repeating ad infinitum with the remaining segments.

It is sometimes known as Cantor dust on account of the dust of points that remain after repeatedly removing the middle thirds. (Cantor dust also refers to the multi-dimensional version of the Cantor set.)

The set has some fascinating, counter-intuitive properties:

  • It is uncountable. That is, there are as many points left behind as there were to begin with.
  • It’s self-similar, meaning each subset looks like the whole set.
  • It’s fractal with a dimension that is not an integer.
  • It has an infinite number of points but a total length of 0.

Wow!

How To Draw The Cantor Set In Google Sheets

To be clear, the Cantor set is the set of numbers that remain after removing the middle third an infinite number of times. That’s hard to comprehend, let alone do in a Google Sheet 😉

But we can create a picture representation of the Cantor set by repeating the algorithm ten times, as shown in this tutorial:

Create The Data

Step 1:

In a blank sheet called “Data”, type the number “1” into cell A1.

Step 2:

In cell B1, type this formula:

={ FILTER(A1:A,A1:A<>"") ;
SUM(FILTER(A1:A,A1:A<>"")) ;
FILTER(A1:A,A1:A<>"") }

Step 3:

Drag this across your sheet up to column J, which creates the data for the first 10 iterations.

Each formula references the column to the left. For example, the formula in cell D will reference column C.

Your data will look like this:

Data For Cantor Dust in Google Sheets

How does this formula work?

It combines array literals and the FILTER function.

Let’s break it down, using the onion framework.

The innermost formula is:

=FILTER(A1:A,A1:A<>"")

This formula grabs all the data from column A and returns any non-blank entries, in this case just the value “1”.

Now we combine two of these together with array literals:

={ FILTER(A1:A,A1:A<>"") ;
FILTER(A1:A,A1:A<>"") }

Here the array literals { ... ; ... } stack these two ranges.

In this first example, it puts the number “1” with another “1” beneath it in column B.

Then we add a third FILTER and also SUM the middle FILTER range to create our final Cantor set algorithm:

={ FILTER(A1:A,A1:A<>"") ;
SUM(FILTER(A1:A,A1:A<>"")) ;
FILTER(A1:A,A1:A<>"") }

As we drag this formula to adjacent columns, the relative column references will change so that it always references the preceding column.

In column B, the output is:

1,1,1

Then in column C, we get:

1,1,1,3,1,1,1

And in column D:

1,1,1,3,1,1,1,9,1,1,1,3,1,1,1

etc.

This data is used in the sparkline to generate the correct gaps for the Cantor set.

Draw The Cantor Set

We’ll use sparklines to draw the Cantor set in Google Sheets.

Cantor Dust In Google Sheet
Click to enlarge

Step 4:

Create a new blank sheet and call it “Cantor Set”.

Step 5:

Next, create a label in column A to show what iteration we’re on.

Put this formula in cell A1 and copy down the column to row 10:

="Cantor Set "&ROW()

This creates a string, e.g. “Cantor Set 1”, where the number is equal to the row number we’re on.

Step 6:

The next step is to dynamically generate the range reference. As we drag our formula down column B, we want this formula to travel across the row in the “Data” tab to get the correct data for this iteration of the Cantor set.

Start by generating the row number for each row with this formula in cell B1 and copy down the column:

=ROW()

(I set up my sheet with the data in columns because it’s easier to create and read that way. But then I want the Cantor set in a column too, hence why I need to do this step.)

Step 7:

Use the row number to generate the corresponding column letter with this formula in cell C1 and copy down the column:

=ADDRESS(1,ROW(),4)

This uses the ADDRESS function to return the cell reference as a string.

Step 8:

Remove the row number with this formula in cell D1 and copy down the column:

=SUBSTITUTE(ADDRESS(1,ROW(),4),"1","")

Step 9:

Combine these two references to create an open-ended range reference for the correct column of data in the “Data” sheet.

Put this formula in cell E1 and copy down the column:

="'Data'!"&ADDRESS(1,ROW(),4)&":"&SUBSTITUTE(ADDRESS(1,ROW(),4),"1","")

This returns range references e.g. 'Data'!A1:A

Step 10:

Put this formula in cell F1 and copy down the column:

=INDIRECT("'Data'!"&ADDRESS(1,ROW(),4)&":"&SUBSTITUTE(ADDRESS(1,ROW(),4),"1",""))

This will show #REF! errors: “Array result was not expanded because it would overwrite data in…”

However, don’t worry, these are only temporary as we’ll dump this data into the sparkline formula next.

Step 11:

In column G, create a default sparkline formula:

=SPARKLINE(INDIRECT("'Data'!"&ADDRESS(1,ROW(),4)&":"&SUBSTITUTE(ADDRESS(1,ROW(),4),"1","")))

This shows the default line chart (except for the first row where it shows a #N/A error).

Step 12:

In column H, convert the line chart sparkline to a bar chart sparkline by specifying the charttype in custom options:

=SPARKLINE(INDIRECT("'Data'!"&ADDRESS(1,ROW(),4)&":"&SUBSTITUTE(ADDRESS(1,ROW(),4),"1","")),{"charttype","bar"})

Step 13 (optional):

Finally, in column I, change the colors to a simple black and white scheme, by specifying color1 and color2 inside the sparkline:

=SPARKLINE(INDIRECT("'Data'!"&ADDRESS(1,ROW(),4)&":"&SUBSTITUTE(ADDRESS(1,ROW(),4),"1","")),{"charttype","bar";"color1","black";"color2","white"})
Cantor Set Data Formulas
Click to enlarge

Feel free to delete any working columns once you have finished the formula showing the Cantor set.

Finished Cantor Set In Google Sheets

Here are the first 10 iterations of the algorithm to create the Cantor set:

Cantor Set In Google Sheet
Click to enlarge

Of course, this is a simplified representation of the Cantor set. It’s impossible to create the actual set in a Google Sheet since we can’t perform an infinite number of iterations.

Can I see an example worksheet?

Yes, here you go.

See Also

You might enjoy my other mathematical Google Sheet posts:

PI Function in Google Sheets And Other Fun π Facts

Complex Numbers In Google Sheets

How To Draw The MandelBrot Set In Google Sheets, Using Only Formulas

The FACT Function in Google Sheets (And Why A Shuffled Deck of Cards Is Unique)

Radial Bar Charts in Google Sheets

In this post, I’m going to show you how to create radial bar charts in Google Sheets.

Radial Bar Chart in Google Sheets

They look great and grab your attention, which is important in this era of information overload.

But they should be used sparingly because they’re harder to read than a regular bar chart (because it’s harder to compare the length of the curved bars).

How To Create A Radial Bar Chart In Google Sheets

Let’s begin with the data.

In this example, we’ll create a radial bar chart in Google Sheets with 3 series.

We need a column of values for these 3 series, for example, products with a number of units sold.

Next, we need some upper limit (max value) for our bars. This allows us to scale the bars properly.

Lastly, we need a helper column that calculates the difference between the max value and the actual value.

Here’s the data for the radial bar chart, in cells E3:H6:

Radial bar chart data table with Formula

Ok, I’m going to let you in on a little secret now…

This is not a single chart. No sir, it’s three charts overlaid on top of each other.

And yes, this means it takes three times as long to create!

Step 1: Create the inner circle

Highlight the first row of data but exclude the max value column. In the example dataset above, highlight E4:G4 and insert a chart.

Select a doughnut chart.

Under the Setup menu, make sure to check the “Switch rows/columns” checkbox, so your chart looks like this:

Donut chart in Google Sheets

Under the customize menu of the chart tool, set the following conditions:

  • Background color: None
  • Chart border color: None
  • Donut hole size: 67%
  • Set Slice 2 color to none
  • Remove the chart title
  • Set the legend to none

This is what the inner donut should look like:

Donut chart in Google Sheets

Step 2: Create the middle circle

Repeat the steps above for the inner circle, but use the next row of data, choose a different color, and set the donut hole size to 77% (you may have to experiment with these percentages to line everything up at the end).

Drag the second donut chart on top of the first and line up the radial bars to get:

Radial Bar Chart in Google Sheets

Step 3: Create the outer circle

Again, repeat the steps above from the inner circle to create a third donut chart, using the third row of data, a different color, and setting the donut hole size to 81% (again, this might need tweaking to line everything up).

Drag this third donut chart on top of the other two and you have a radial bar chart in Google Sheets!

Radial Bar Chart in Google Sheets

Note on editing charts:

Since the charts are placed on top of each other, you’ll only be able to access the top chart to edit. You’ll have to move it to the side to access the chart underneath, and then move that one if you want to access the inner chart.

Step 4: Add the data labels

It gets messy to add the data labels to each chart through the chart editor, so I opted to create formulas to add my data labels into the cells next to each bar of the radial bar chart.

To access cells underneath the charts, click on a cell outside of the chart area and then use the arrow keys on your keyboard to reach the desired cell.

Once there, add the following formula:

=E6&": "&TEXT(F6,"#,0")

This formula uses the TEXT function to combine text and numbers in Google Sheets.

This shows the series name and value alongside each bar:

Data labels for radial bar chart in Google Sheets

To finish, remove the gridlines from your Sheet to give the chart a clean look.

Radial Bar Chart in Google Sheets

Can I see an example worksheet for the radial bar chart?

Yes, here you go.

Real World Examples of Radial Bar Charts

My friend Jeff Sauer, who founded Data Driven U to teach people data-driven marketing, contacted me recently about creating a radial bar chart for one of his workshops.

He is graciously sharing his report here, so you can see a radial bar chart with six rings:

Data Driven U chart

This is a screenshot of his Google Sheet!

(If you’re looking for top draw digital marketing, then you should definitely check out Jeff’s site: DataDrivenU.com This is not an affiliate link, just a personal recommendation!)

You’ve probably also seen a radial bar chart in the wild with the Apple Watch Rings Chart!

Apple Watch Ring Chart

How to Create an Annotated Line Graph in Google Sheets

line graph in Google Sheets
Animated line chart in Google Sheets

This post looks at how to make a line graph in Google Sheets, an advanced one with comparison lines and annotations, so the viewer can absorb the maximum amount of insight from a single chart.

For fun, I’ll also show you how to animate this line graph in Google Sheets.

Want your own copy of this line graph?

Click here to access your copy of this template >>

This chart was originally developed for The Write Life during their 4-day product sale earlier this year. It featured as part of a dashboard that was linked to the E-junkie sales platform and displayed sales data in real-time:

Google Sheet e-junkie real-time dashboard
Google Sheet e-junkie real-time dashboard

As with any graph, we start with the data:

The data table

Line graph data table

The key to this line graph in Google Sheets is setting up the data table correctly, as this allows you to show an original data series (the grey lines in the animated GIF image), progress series lines (the colored lines in the animated GIF) and current data values (the data label on the series lines in the GIF).

In this example, I have date and times as my row headings, as I’m measuring data across a 4-day period, and sales category figures as column headings, as follows:

Annotated line graph data table

Red columns

The red column, labeled with 1 above, contains historic data from the 2015 sale.

Red column 2 is a copy of the same data but only showing the progress up to a specific point in time.

In red column 3, the following formula will create a copy of the last value in column 2, which is used to add a value label on the chart:

=IF(AND((C2+C3)=C2,C2<>0),C2,"")

Purple columns:

Purple columns 4,5 and 6 are exactly the same but for 2016 data. The formula in this case, in column 6, is:

=IF(AND((F2+F3)=F2,F2<>0),F2,"")

Green columns:

Data in green columns 7 and 8, is our current year data (2017), so in this case there is no column of historic data. The formula in column 8 for this example is:

=IF(AND((H2+H3)=H2,H2<>0),H2,"")

Creating the line graph in Google Sheets

Highlight your whole data table (Ctrl + A if you’re on a PC, or Cmd + A if you’re on a Mac) and select Insert > Chart from the menu.

In the Recommendations tab, you’ll see the line graph we’re after in the top-right of the selection. It shows the different lines and data points, so all that’s left to do is some formatting.

Line graph selection

Format the series lines as follows:

  • For the historic data (columns 1 and 4 in the data table), make light grey and 1px thick
  • For the current data (columns 2, 5 and 7 in the data table), choose colors and make 2px thick
  • For the “max” values (columns 3, 6 and 8 in the data table), match the current data colors, make the data point 7px and add data label values (see steps 1, 2 and 3 in the image below)

Line graph data labels

This is the same technique I’ve written about in more detail in this post:

How can I annotate data points in Google Sheets charts?

Animating the chart with Apps Script

How about creating an animated version of this chart?

Oh, go on then.

When this script runs, it collects the historic data, then adds that data back to each new row after a 10 millisecond delay (achieved with the Utilities.sleep method and the SpreadsheetApp.flush method to apply all pending changes).

I don’t make any changes to the graph or create any fancy script to change it, I leave that up to the Google Chart Tool. It just does its best to keep up with the changing data, although as you can see from the GIF at the top of this post, it’s not silky smooth.

By the way, you can create and modify charts with Apps Script (see this waterfall chart example, or this funnel chart example) or with the Google Chart API (see this animated temperature chart). This may well be a better route to explore to get a smoother animation, but I haven’t tried yet…

Here’s the script:

function startTimedData() {
  var ss = SpreadsheetApp.getActive();
  var sheet = ss.getSheetByName('Animated Chart');
  var lastRow = sheet.getLastRow()-12;
  
  var data2015 = sheet.getRange(13,2,lastRow,1).getValues(); // historic data
  var data2016 = sheet.getRange(13,5,lastRow,1).getValues(); // historic data
  
  // new data that would be inputted into the sheet manually or from API
  var data2017 = [[1],[7],[14],[19],[27],[32],[34],[36],[44],[49],[57],[65],[72],[76],[79],[86],[92],[99],[104],[109],[111],[112],[120],[128],[130],
                  [132],[133],[140],[144],[149],[151],[152],[158],[162],[170],[177],[179],[184],[188],[194],[200],[205],[211],[216],[224],[232],[238],
                  [241],[246],[248],[252],[259],[266],[268],[276],[284],[291],[299],[300],[301],[306],[311],[315],[316],[323],[324]];
  
  for (var i = 0; i < data2015.length;i++) {
    outputData(data2015[i],data2016[i],data2017[i],i);
  }
  
}

function outputData(d1,d2,d3,i) {
  var ss = SpreadsheetApp.getActive();
  var sheet = ss.getSheetByName('Animated Chart');
  
  sheet.getRange(13+i,3).setValue(d1);
  sheet.getRange(13+i,6).setValue(d2);
  sheet.getRange(13+i,8).setValue(d3);
  Utilities.sleep(10);
  SpreadsheetApp.flush();
}

function clearData() {
  var ss = SpreadsheetApp.getActive();
  var sheet = ss.getSheetByName('Animated Chart');
  var lastRow = sheet.getLastRow()-12;
  
  sheet.getRange(13,3,lastRow,1).clear();
  sheet.getRange(13,6,lastRow,1).clear();
  sheet.getRange(13,8,lastRow,1).clear();
  
}

On lines 6 and 7, the script grabs the historic data for 2015 and 2016 respectively. For the contemporary 2017 data, I’ve created an array in my script to hold those values, since they don’t exist in my spreadsheet table.

This code is available here on GitHub.

Finally, add a menu for access from your Google Sheet with the following code:

function onOpen() {
  var ui = SpreadsheetApp.getUi();
  
  ui.createMenu("Timed data")
    .addItem("Start","startTimedData")
    .addItem("Clear","clearData")
    .addToUi();
}

This allows you to run the Start and Clear functions directly from your Google Sheet browser tab, rather than the script editor tab.

That’s it. Hit Start and you should see your chart animate before your eyes:

Animated Apps Script chartAnimated Apps Script chart

If you look closely, you’ll also see the data populating your sheet.

Funnel charts in Google Sheets using the chart tool, formulas and Apps Script

Let’s talk about funnel charts in Google Sheets.

The charts themselves are a bit of a novelty. Yes, they’re aesthetically pleasing because of that resemblance to a real-world, tapering funnel, which reinforces their message, but a plain ole’ bar chart would be equally suitable and actually easier to read data from (because the bars have a common baseline).

However, they throw up some interesting techniques in Google Sheets and for that reason, merit this long article.

We’ll build them using tricks with the chart builder tool, then with two different types of funky formula and finally, and best of all, we’ll build a tool using Apps Script, as shown in this image:

Funnel charts with apps script in Google Sheets

As with the waterfall charts in Google Sheets, they’re not one of the out-the-box charts available to us, so we have to manually create them with a crafty workaround. Thankfully, they’re relatively simple to create, certainly simpler than the waterfall chart.

For all of these examples, we’ll use this fictitious real-estate dataset:

Google Sheets funnel chart data

Here, I’m imagining the real estate agency collects data relating to their sales funnel, and they want to display it in a funnel chart format.

Click here to open up the Google Sheet template and make your own copy (File > Make a copy...).

Continue reading Funnel charts in Google Sheets using the chart tool, formulas and Apps Script

How to create a waterfall chart in Google Sheets

Update December 2017: Google have recently added Waterfall Charts to the native charts in the Chart Tool of Google Sheets, obviating the need for you to manually create your waterfall charts (or use apps script) per my original post.

Now you simply highlight your data, click Insert > Chart and under the Chart type picker choose “waterfall”, as shown in the following image:

New native waterfall charts in Google Sheets


The original post that follows was first published in late 2016, and I’m leaving it here for anyone who wants to look under the hood at how waterfall chart data is constructed and how to do that using apps script.


Original article:

In this post, we’ll look at how to create a waterfall chart in Google Sheets.

Waterfall charts are real. And useful. They show the cumulative effect of a series of positive and/or negative values on an initial starting value.

The following waterfall chart shows the headcount changes for a department, visually depicting the cumulative effect of the additions and deletions to the start value:

Headcount Waterfall Chart

It shows the number of staff in our department at the start of the year (left grey bar), the number of people added from other departments or as new hires (green bars), the number of people who left (red bars) and finally the balance which is the headcount at the end of the year (right grey bar).

The waterfall chart above is relatively easy to create in Google Sheets but does still require some data wrangling to set it up. Notice that all of the bars are above the x-axis (Case 1), which makes the data set up vastly simpler than the case when we have a mix of bars above and below the x-axis, or spanning the x-axis (see Case 2 below).

I’ll show you how to create both of these cases, starting with the easier, positive-bar case.

After creating the simple and complex versions manually with formulas, I’ll show you some Apps Script code to automate the majority of the process and massively speed up creating complex waterfall charts.

Templates are available for all three methods, with links at the end of each section and at the end of this post.
Continue reading How to create a waterfall chart in Google Sheets