# Why are the wet patches on these floor tiles circular?

My friend's 3-year-old daughter asked "Why are there circles there?"

It had either rained the night before or frost had thawed. What explains the circles?

Follow-up question: Ideally, are these really circles or some kind of superellipse?

• Are the tiles made from rubber? Are they convex?
– xvk3
Mar 4 '18 at 7:18
• I believe they are flat, stained concrete blocks. Mar 4 '18 at 21:58
• Mar 5 '18 at 11:23
• Funny enough, I saw the opposite pattern on some tiles the other day. The grout and surroundings were wet while the middle was dry.
– JMac
Mar 5 '18 at 19:20
• Some good answer here but remember, She's three. Tell her it is because the edges dry quicker than the middle. The chances are that she will accept this and this will be enough. If she asks more, then go on to explain why.
– josh
Mar 6 '18 at 9:18

Both thawing and evaporation involve heat exchange between the stone tile, the water sitting atop the stone tile, any water that's been absorbed by the stone tile, and the air around. The basic reason that the center and the edges of the tile evaporate differently is that the gaps between the tiles change the way that heat is exchanged there. However the details of how that works are a little more involved than I can get into at the moment, and would be lost on a three-year-old anyway.

A good way to explain this phenomenon to a three-year-old would be to bake a batch of brownies in a square pan, and watch how the brownies get done from the outside of the pan inwards. Even after you have finished them you can still tell the difference between the super-crispy corner brownies, the medium-crispy edge brownies, and the gooey middle-of-the-pan brownies. The three-year-old would probably ask you to repeat this explanation many times.

I think the shapes are not exactly circles, superellipses, or any other simple mathematical object --- there's too much real life in the way --- but they do become more circular as the remaining puddle gets further from the edges.

• If we assume it is dominated by the diffusion equation and starts from a square initial condition one can probably make an analytic expression for the shape. But explaining boundary value problems to 3-year olds is a tricky. Mar 3 '18 at 23:17
• +1 for including a demonstrative and practical yet tasty experiment for kids of all ages (including mine). Mar 4 '18 at 0:12
• "However the details of how that works are a little more involved than I can get into at the moment" -- honestly that sounds like a bit of a non-explanation. There must be a reason for the temperature to be uneven, and if we don't know what that is, we can't really explain the pattern. (I suspect it's because the gout between the tiles absorbs more heat from sunlight because it's darker, but that's just conjecture.) Mar 4 '18 at 0:31
• @Nathaniel I came up with some arguments for the edges being warmer and some arguments for the edges being cooler, depending on some details of the construction that I don't have access to and that I don't find interesting. The point is just that heat flow near the edges and near the middle are different.
– rob
Mar 4 '18 at 0:35
• I guess the only thing I would say in commentary is that children, even those less than five years old, deserve better than to presume that they couldn't understand something. Better that you make the attempt and they don't than you don't and they could've. Mar 5 '18 at 17:12

According to @rob , "The basic reason that the center and the edges of the tile evaporate differently is that the gaps between the tiles change the way that heat is exchanged there. " This sounds reasonable. However, I'd like to offer some other tentative explanations. First of all, let me note that the tiles in the image look unglazed and, therefore, their surface is probably porous. So there maybe two additional explanations:

1) Pore distribution may depend on the distance from the edges due to peculiarities of the tile manufacturing process;

2) Tiles dry out faster near the edges because initially water there partially flows away to the gaps between the tiles, as the surface of the gaps is typically somewhat lower than the surface of the tiles.

EDIT (03/06/2018): @Yly and @alseether believe that this approach does not explain the difference between the water spots on the "grey and red" tiles. It does not, indeed, so I wrote earlier in a comment to rob's answer: "First of all, porosity (or some other properties) of the surface of the "grey and red" tiles can be different, but, for the sake of the argument, let us assume that they only differ in color (and, therefore, albedo). So the grey tiles can dry faster, but this mechanism can work in parallel with the porosity/runoff mechanism."

• The way the tiles are grouted into place may also lead to greater mobility and wear on the centers.
– arp
Mar 4 '18 at 21:54
• I like this answer, but I think it misses a peculiar point (@rob's answer misses too). Why red tiles evaporate less than geen-ish ones? Different materials? Less porous? Mar 6 '18 at 12:46
• @alseether How do you know the red tiles evaporate less? We're not seeing water directly, we're seeing color change from water. Perhaps the red tiles exhibit a color-changing response to water that is different from that or green ones. Mar 6 '18 at 16:38
• I would posit #2 is the answer. Green tiles appear to have a different pore size and distribution Mar 7 '18 at 19:10
• I recall reading an article in scientific american about a theory for why plants are green that noted that blue light highly energetic but low intensity and red light is lower energy but high intensity in sunlight. By rejecting green light, the plants were optimizing towards the most energy. If this is correct, then that would imply that the green tiles are going to absorb more solar energy than the red tiles. This could explain by the green tiles dry more quickly. Mar 9 '18 at 22:15

The sides and corners of the cube are a few degrees warmer than the center.

They are there for the same reason that an ice-cube's corners go round, and the same reason that square stones lose their corners first in the river.

They aren't precisely circles, they are rounded squares also known as squircles. What would a child think of that word?

You can see from the shadow that the sun has warmed the upper left sides of the squares the fastest.

The corners and edges warm up first, especially the ones that catch the sunshine. The dark tiles are also hotter. (irradiance incidence and mathematics in W/m2 and mJ/m2 can perhaps be simplified into child concepts)

The water may also act as a solar reflector and reduce the surface area of the stone. Warmth from the ground may be escaping through the sides too.

The rain may have gathered more deeply towards the middle of the cube too, due to runoff dynamics and capillary action. Without sunshine, the capillary action would also thin the water from the edges and the corners faster than from the center.

Information for adults rather than children: solar irradiation in W/m2, Paris is 47'N and Moscow is 55'N.

• What cube, I see a square... I don't understand how you justify the sides being several degrees warmer or that they would heat up faster, or even get enough sunlight to account for the difference.
– JMac
Mar 5 '18 at 19:24
• My guess would be that one edge drying quicker is due to prevailing wind/direction of airflow, not sunlight.
– josh
Mar 6 '18 at 9:13
• Cool comments, I thought you had me pegged briefly. However, the dark tiles have evaporated a lot faster than the red ones, which confirms that it is the sunlight. The atmospheric to terrestrial energy absorption ratio from the sun depends on atmospheric clarity and time of day, the stones of average albedo becomes the primary energy absorber very early in the morning. The average day of irradiance from the sun in Moscow is over 150W/m2 on a clear day at this time of year. I wish I could research irradiance W/m2 and thermal inertia of cement, the child wouldn't comprehend. Mar 7 '18 at 14:03
• The sun affecting dark ones more still doesn't really explain the sides being warmer. Your description of how the sun affects edges more seems like a poor analogy for sunlight; because sunlight is only really hitting one surface; while ice cubes and stones in rivers experience the effects on all surfaces.
– JMac
Mar 7 '18 at 14:10
• The permeability and capillary action of the cement keeps most dampness to the center, which is a notable element of the physics of the tiles. You bring that to my attention. You say the wind is from the NW. The wind chill factor that you speak of is not clearly visible from the image compared to the irradiance, which has provided the black tiles with a lot more energy. The irradiance is a very dominant energy source as demonstrated by the dark tiles, and chamfers and edges of the tiles are facing the sun, especially on the left towards the morning sun. ony way to check is photo a shade place Mar 7 '18 at 14:37

Others have emphasised the effects of evaporation, but I'd suggest another key factor is likely to be related to several features of the grouting that make the edges dry sooner.

Firstly, the fact that the grout sits at a lower level from the top of the tile means moisture will tend to migrate to the lower level - and it will "drain" from the edges of the tile sooner than it drains from the middle of the tile.

Secondly, the grouting tends to be more porous than tile, and will absorb moisture more easily and conduct it away to drier areas due to capilliary action.

Thirdly, if the floor underneath is not sealed, then moisture will tend to migrate through the grout more quickly into the base and seep away, whereas the tile face is typically moisture-proof.

Another possibility is that there may be slight manufacturing imperfections in the planar surface of the tile, with a slight depression in their centre, which will tend to retain moisture against all the above factors.

I would suggest an alternative explanation: the tiles are not, in fact, perfectly flat.

"Flat" floor tiles actually have a degree of acceptable curvature when manufactured (similar to this list of deviations) which is small enough that it's not visible, and - as you will have doubtless observed in older tiles - tend to wear more at the centre over time, making them more concave. It is likely therefore that there is a small degree of curvature to the tiles you observed. This curvature is not marked enough to be directly visible, but enough that water will pool slightly more in the centre and run down the tile towards the centre as it evaporates from the edge.

While there are likely also thermodynamic differences, I think this is likely to have a bigger effect.

• This should be easy to verify placing a ruler on top of it. Mar 8 '18 at 9:47

The grout betwen the tiles is dark, and has a rougher surface than the tile. (It's composed of sand, cement, and colorings.) It's also slightly lower than the tile surface. Therefore it collects solar heat better than the lighter colored tiles, and as the heat is conducted inwards to the tile, the frost melts/water evaporates.

• I'm not sure why every other answer here doesn't note the grout being black. Mar 5 '18 at 14:37
• This could be tested by placing a strip of white tape on top of the grouting, on one-side of a tile, and observing the difference. Mar 9 '18 at 13:36

Let's first observe the phenomenon more closely:

• there is less or no water on the green tiles. It seems the brownish/reddish tiles have dried slower.
• the perimeter of all tiles has dried faster than the centre.
• the filling in the perimeter of the tiles (which is also between tiles) is darker than the tiles. In fact, the further a location is from the perimeter, the less chance that it is dry.
• since frost was mentioned, it is probably winter or a cool time of the year. That is, the air is colder than ground temperature (ground temperature meaning at $1m$ depth and below).

Possible solutions:

Explaining the colour differential:

• Darker colours absorb more wavelengths of light and heat, so they will get warmer under the same light or the same visible/near visible radiation. Some insights from this stackexchange and a good students' experiment on melting ice cubes on different colours here.
• Accordingly, the green tiles are darker than the brownish reddish tiles, so they absorb more heat, making the water above them warmer, giving water molecules at the surface more energy, thus allowing them to escape in larger droves within a given time. The greener tiles thus dry faster.
• The water on both types of tiles is equally reflective and transparent to light, so no difference expected from that aspect.

Explaining the perimeter/centre differential:

• The filler material between the tiles looks darker and thus absorbs more light and heat, so that water at the perimeters evaporates more quickly.
• The effect of the warmer perimeter diminishes as you move away from the line of the filler material between the tiles.

and furthermore...

• Whilst the effect of colour is evidently stronger here, ground heat effect is likely not neutral. The ground is a large heat sink, so that parts of buildings that are at ground level or below ground level are less affected by air temperature variation during the year, tending to be more temperature-stable. This phenomenon makes geothermal heat pumps practical alternatives to air conditioning using the refrigeration cycle.
• If the tiles conduct heat better than the perimeter filler material, then the ground heat effect is reducing the (surface colour-driven) contrast between the perimeter and the centre of the tiles since more heat is conducted upwards through the centre of the tiles than through the perimeter.
• If the filler material is a better conductor of heat however, the ground heat effect exaggerates the phenomenon of 'round' patches to some degree.

In the most straight forward and basic terms..

Water that resides close to a side face of the tile (near the cracks) will exit the tile in both a horizontal and vertical fashion, thus allowing a quicker rate of drying within those regions of the tile's volume. The water in the middle, however, only leaves the tile vertically.

And then, as @rob mentioned, neighboring tiles also assist in heating each other, so, there's a kind of compounding effect when considering multiple and closely adjacent tiles; or, at least, initially there is.

Lastly, keep in mind that the color of the tile also plays a major role in the overall evaporation rate. As you can see, green tiles have evaporated most quickly, with red lagging behind, and this is to be expected. If these tiles weren't uniform in color, and, oriented in such an alternating fashion, you probably wouldn't have observed such circular shapes for the remaining moisture.

Analogy and demonstration (in 2-D):

Imagine there are three checkout lines at a grocery store (left, middle, right). Each line has the same number of people initially.

For the middle line, customers can only check out using the counter that exists in the very front of the line. For the left and right lines though, with each step forward, people will also be able to use checkout counters that exist along the sides of the check out lines, and not just at the front.

Depending on the number of checkout counters that exist along the sides of each line, the left and right lines will diminish that much quicker than the middle line, eventually leaving just the middle checkout line.

This could very easily be demonstrated with a small group of people, or, a handful of toys / action figures.

Have you checked whether the tiles are really completely flat? The easiest explanation would be that in the middle of the tiles the water is a little deeper than at the outside and that the water evaporation starting in the shallower outside parts causes these quasi-circular water puddles. You could easily test this by pouring some water on a dry tile and then see whether a puddle forms in the middle.