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Consider the lowly wall clock. It's practical and generally tells the correct time. It's the same clock everywhere and after a while it gets boring pretty quickly—maybe now?

In the regular clock the face bezels stay in place and the hands move. Why am I telling you this? Well, maybe you see where I'm going.

Who says it's the hands that have to rotate? Instead of rotating hands and a stationary bezel, consider the clock with stationary hands rotating bezels.

In the Ptolemaic clock there are two independent bezels and two independent hands. The **bezels rotate counterclockwise** to simulate the standard clockwise motion of the hands. The hands are not moving but in the frame of reference of the bezels, it's the hands that are rotating. The position of the bezel is always related to the current time and the position of its corresponding hand.

The bezel can move clockwise.

Thanks to Rodrigo Goya for suggesting the name for this kind of clock—Ptolemaic Clock, named so after the geocentric Ptolemaic model of the solar system.

To tell the time on the Ptolemaic clock is a process identical to using the standard clock. You look at the bezel numbers at the ends of the hour and minute hands.

On the fixed bezel layout, most people will take a short cut and tell the time by the position of the hands. This works as long as you have a standard clock. On a Ptolemaic clock the position of the hands tells you nothing.

Here is a Ptolemaic clock telling us it is 6:30. It uses the same position of hands as in the figures above.

You know this because the blue hour hand points to midway between 6 and 7 on the inner hour bezel and the grey minute hand points to 30 on the outer minute bezel.

After 15 minutes, it's 6:45 and our Ptolemaic clock bezels have moved a little bit.

Can you tell what time it is on the Ptolemaic clock below?

Customizing your Ptolemaic clock is easy. Simply adjust the hands to desired positions and set the time by moving the bezels. The clock below shows the same time as the clock in the above figure — both show 8:50.

In the clock design shown here, the hands are the same size and only differ by color. To make things less confusing, emphasize the hour hand.

To make things more confusing, remove all color and number cues, keeping only a single symbol on each of the bezels to indicate 12 o'clock and 0 minutes. This is shown in the clock below.

Spice it up with multiple Ptolemaic clocks side-by-side telling the same time with different hand positions.

Suppose it is 2:30 in Vancouver—this is my location. The clocks below all show 2:30, but with hands set to 5:30, 11:30 and 7:30.

These hand positions are those that would appear on a standard clock showing the times in New York (5:30), Paris (11:30) and Tokyo (7:30).

Let's now use the Ptolemaic clock to show times at these three locations but with the hand set to the curiously satisfying layout of 10ish minutes to 2.

Set both hand positions to 12 o'clock and then remove the hands; to tell time, read the numbers on the hour and minute bezels at the apex of the clock.

Sophisticated implementations of the Ptolemaic clock could periodically randomize hand positions to keep things interesting; by the time you've figured out the time in the morning, you're wide awake.

Every minute the clock randomly resets its hand positions. The movement is smooth and the bezels follow.

If you would like to implement the Ptolemaic clock, I would be happy to hear from you. One should be able to take a regular wall clock, reverse the direction of the hand mechanism and rig a freely moving bezel to each of the minute and hour mechanism. The hands should not move and can be fixed to the front glass plate, for example.

It should now be clear that the Ptolemaic clock is superior to the standard clock. The reasons are

- it's much harder to tell time on the Ptolemaic clock, which makes your brain do more work
- it tips its hat off to a simpler time when we didn't know anything and hints at the possibility of regression anytime
- it will confuse everyone
- you have a great excuse for being late
- return to geocentric values!

- you can customize your own Ptolemaic clock by moving the hands to arbitrary locations
- two Ptolemaic clocks can have their hands and bezels at different positions but still be telling the same time
- two Ptolemaic clocks can have their hands at the same position but be telling different times

We discuss the many ways in which analysis can be confounded when data has a large number of dimensions (variables). Collectively, these are called the "curses of dimensionality".

Some of these are unintuitive, such as the fact that the volume of the hypersphere increases and then shrinks beyond about 7 dimensions, while the volume of the hypercube always increases. This means that high-dimensional space is "mostly corners" and the distance between points increases greatly with dimension. This has consequences on correlation and classification.

Altman, N. & Krzywinski, M. (2018) Points of significance: Curse(s) of dimensionality *Nature Methods* **15**:399–400.

Inference creates a mathematical model of the datageneration process to formalize understanding or test a hypothesis about how the system behaves. Prediction aims at forecasting unobserved outcomes or future behavior. Typically we want to do both and know how biological processes work and what will happen next. Inference and ML are complementary in pointing us to biologically meaningful conclusions.

Statistics asks us to choose a model that incorporates our knowledge of the system, and ML requires us to choose a predictive algorithm by relying on its empirical capabilities. Justification for an inference model typically rests on whether we feel it adequately captures the essence of the system. The choice of pattern-learning algorithms often depends on measures of past performance in similar scenarios.

Bzdok, D., Krzywinski, M. & Altman, N. (2018) Points of Significance: Statistics vs machine learning. Nature Methods 15:233–234.

Bzdok, D., Krzywinski, M. & Altman, N. (2017) Points of Significance: Machine learning: a primer. Nature Methods 14:1119–1120.

Bzdok, D., Krzywinski, M. & Altman, N. (2017) Points of Significance: Machine learning: supervised methods. Nature Methods 15:5–6.

Celebrate `\pi` Day (March 14th) and go to brand new places. Together with Jake Lever, this year we shrink the world and play with road maps.

Streets are seamlessly streets from across the world. Finally, a halva shop on the same block!

Intriguing and personal patterns of urban development for each city appear in the Boonies, Burbs and Boutiques series.

No color—just lines. Lines from Marrakesh, Prague, Istanbul, Nice and other destinations for the mind and the heart.

The art is featured in the Pi City on the Scientific American SA Visual blog.

Check out art from previous years: 2013 `\pi` Day and 2014 `\pi` Day, 2015 `\pi` Day, 2016 `\pi` Day and 2017 `\pi` Day.

We examine two very common supervised machine learning methods: linear support vector machines (SVM) and k-nearest neighbors (kNN).

SVM is often less computationally demanding than kNN and is easier to interpret, but it can identify only a limited set of patterns. On the other hand, kNN can find very complex patterns, but its output is more challenging to interpret.

We illustrate SVM using a data set in which points fall into two categories, which are separated in SVM by a straight line "margin". SVM can be tuned using a parameter that influences the width and location of the margin, permitting points to fall within the margin or on the wrong side of the margin. We then show how kNN relaxes explicit boundary definitions, such as the straight line in SVM, and how kNN too can be tuned to create more robust classification.

Bzdok, D., Krzywinski, M. & Altman, N. (2018) Points of Significance: Machine learning: a primer. Nature Methods 15:5–6.

Bzdok, D., Krzywinski, M. & Altman, N. (2017) Points of Significance: Machine learning: a primer. Nature Methods 14:1119–1120.

In a Nature graphics blog article, I present my process behind designing the stark black-and-white Nature 10 cover.

Nature 10, 18 December 2017

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