Data Visualization

Visualizing Time with the Infinity Hour Chart

This is another experiment in visualizing 24-hour cyclical data. My last post explored a method of linear representation (the Double Time Bar Chart). Linear representations have problems when it comes to showing the cyclical nature of time data (ie there is no start or end of a 24 hour cycle).


When trying to think of visual representations of never-ending cycles I was inspired by the infinity symbol. It’s a great symbol to show a continuous cycle, while at the same time being more visually interesting than a simple circle (fun fact: the infinity symbol dates back to 1655). The other iconography that came to mind when thinking about infinity is the hour glass. An hour glass not only represents time, but it also looks similar to a vertical infinity symbol.

My thought was that maybe I could combine the two to create a vertical infinity symbol that evokes the metaphor of an hour glass.

Back of the napkin

My original sketch of this concept was done on the back of a napkin. This is the first sketch, which shows how I was originally working with a horizontal infinity symbol.

I experimented with a few different options for how to show the data using fills. One of the sketches (if turned vertically) looks like an hour glass filling up with water on the bottom, reminiscent of the Wikileaks logo.

Drawing Infinity

The mathematical name for the infinity symbol is lemniscate, and more specifically the lemniscate of Bernoulli. With some good Googling you can find algorithms to draw the lemniscate of Bernoulli, which is what I did.

To start I divided the lemniscate into 24 segments, one for each of the hours of the day. My initial plot of the lemniscate in 24 parts looked like this:

I mapped the hours of the day onto this form, with 12pm noon at the very top and the infinity symbol crossing itself at 6pm/6am.

You follow the time by working your way around the infinity. If you start at the top of the symbol at noon, you would start moving around clockwise to 1pm, then 2pm, etc. You’ll reach the center at 6pm, at which point the symbol crosses itself and you then read it counter-clockwise around the bottom.

What you end up with is a way of dividing up the times of day into quadrants. The top-left quarter of the image is the morning, from 6am-12pm. Then the top-right is the afternoon, from 12pm-6pm. Then you have the evening in the botom left (6pm-midnight) and then late-night is in the bottom-right (12am-6am). These quarters match well with how I mentally categorize times of day.

Because the form crosses over itself you can actually read the chart almost in a left-to-right way for both the day (top) and night (bottom).

Drawing Data

The next step is to try to use this form to represent real data. Here’s an example that shows the distribution of driving under the influence arrests in San Francisco:

We can see that this particular crime is primarily a night-time activity that surges around midnight and starts falling off after about 2am. I’ve colored the range of 6am-6pm in orange to show day-time and the range of 6pm-6am in blue to indicate night-time.

For comparison here’s apartment burglary, which is mostly a day-time activity:

Once the viewer understands how to read the chart we can remove the labels and simply show the pattern. Here’s a comparison of a few night-time crimes:

Here’s a comparison of different types of burglary, some of which occur mostly during the daytime (residential burglary) and some of which occur in the afternoon and late at night (burglary of a store).

Small Multiples

Here’s a final example of many different crime types represented side by side to try to see how this chart works for comparisons.


I’m not very happy with this chart in terms of the viewer’s ability to accurately read the chart. I also don’t think it highlights changes between hours enough. Often there are changes and trends that are easy to spot in the linear charts of my last post, but that are very difficult to see in these charts. Each hour is at quite a different angle than the hours on either side, which makes it difficult to compare two hours. You still get the big picture trends, like if a crime is a night-time or day-time crime, but the smaller trends are much harder to spot.

On the flip side, I really like the metaphors of the infinity symbol and the hourglass. On an artistic and philosophical level I think those metaphors make this a really beautiful visualization. Too bad it’s not also effective 🙂


17 thoughts on “Visualizing Time with the Infinity Hour Chart

  1. I like this! The visualisation of 24 hours in the form of a lemniscate (learnt a new word there) could also be beautiful on a wristwatch (infinity symbol with a moving dot as the single indicator of the time) or as an app for computers / mobile devices.

  2. Anonymous says:

    Why just not cutting a circle in 24 instead of 12?

    Also a square could be interesting witch each side representing 6 hours helping better comparisons.

    Good luck

  3. I love the thinking here. It’s interesting to come up with new ways
    to represent data, especially trying to get back to the properties of the measured
    variable (time being cyclical).
    Most importantly though, you’ve acknowledged that the solution, while novel, doesn’t really work
    in practice or add much value. People too often stop before that critical stage, and we’re left with plots and infographics that are impossible to interpret.

    Keep up the experimenting; I’m interested to see what solution is most easily parsed by humans. There’s obviously a slew of approaches, but it’s not immediately obvious which one is right. The key problem seems to be that we need to visualise the data non-linearly to represent the cyclicality of time, but then we lose the relatively accurate comparability of bar chart bars.


  4. David A R Kemp says:

    As @Anonymous says, surely an analogue clock would make this easier to read? we’re used to following the circle around left-to-right, but the cross-over in the the lemniscate seems to break that intuitive action.

  5. Arun Thampan says:

    I find an interesting connection here: because of the underlying theme of a circle->orbit->ellipse – the projection of earth’s yearly transit around the sun is an infinity shaped path called Analemma.

  6. Very interesting – I’ve used something similar to this to visualize insertion/playback points in a MIDI sequencer, where Time is really of the essence! 🙂 Its nice to see you’ve made a similar discovery..

  7. Philip says:

    Love where you’re going with this. I can get behind a creative, off-the-wall idea like this. Unfortunately, like you said, humans have a hard time visualizing an accurate comparison of the area of two or more polygons, even adjacent ones, and thus the big-picture trends are the only standout features. But your work is moving in the right direction and I’m excited to see more.

  8. ash says:

    I also like the experimentation. Data visualization is all about trying to expose hidden patterns of information in new forms.

    But there’s nothing wrong with a good old bar chart. The earlier comment by Anonymous had some good ideas:

    * a 24 hour circle bar chart – i was thinking the bars should be colored to time of day, but when i think lighter colors, i think positive; darker colors negative. that might not be so good

    * dividing the day up into a square of 6 hour blocks – might be interesting if all the blocks rotated when you selected a radio button for time of day. you could also have a block for each day of the week and see a side-by-side comparison

  9. Sam D says:

    Just a comment to say I love the experiment. I agree with your conclusions – but I *wish* it worked, because the concept and the visual are so appealing… Sitting from my distance away I hope you have the time and energy to continue on this thread.. who knows where it could end up!

  10. komiska says:

    totally agree with Martin Roell! id LOVE to have a wristwatch and an app for my mbp with this as animation. i think it would be easier then to “read” it.
    you’ll understand what i mean if you see this artist’s work:

    “In the bright front she exhibits on the one hand colour photographs which reveal a certain motion in time (just under one second), and on the other black and white stills of discrete moments (a few fractions of a second). What the images have in common is how they reveal the eternal flux of a metamorphosing form, which otherwise (without the motions of the body) would simply remain static.

  11. Richard Williams says:

    Great article and interesting thoughts about cyclical data; however, I do find your ideas flawed, in that cyclical data can’t be compared to infinity loop as any cyclical system–let me explain by using your DUI examples above.

    Of course, as your cyclical chart shows drink driving increases during the night and falls during the day, but you fail to figure the chaotic nature of cyclical systems. For instance on New Year’s day (due to late night drinking) the cyclical chart for that day will be far different to a ‘normal’ day. And there will be many of these exceptional periods, which will skew averages and the shape of any cyclical diagram. (burglaries for instance, rise several fold during vacation seasons etc where daylight robbery becomes more prevalent as people are leave homes empty).

  12. Richard Williams says:

    Also, as an interesting aside, increasingly physics theorists are coming to the conclusion that infinity doesn’t exist (in that nothing in the universe, whether it’s distance, time or the number of atoms is limited and therefore nothing can be considered infinite).

  13. A thought about increasing perceptible difference between adjacent segments…

    It looks like you are using the lemniscate boundary as a baseline and midpoint for each segment. What if the line segments only extended outward from the lemniscate baseline. Would this make the comparison between two segments easier and more visually consistent? It would get closer to the benefits of a traditional bar graph.

    You may also be able to add another data set to the inside of the chart. The biggest challenge would be the center point at which the lines overlap.

    … but just a thought.

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  15. hannah says:

    there is also the issue with “infinity times infinity” If the simple infinity does actually represnt the times of day, but over lapping at 6pm and am, then what is to say for infinity times infinity? and where does the simbol stand in ancient history?

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