Newton who laid out a set of mathematical rules to predict the motion of objects from a falling apple to planets orbiting the Sun, however it was Einstein who described how two masses tend to accelerate towards each other; he formulated the general relativity.

Flatland is a story written by Edwin Abbott showing the amazing and bizarre world in which there are creatures living in a world with only two spatial dimensions and one time dimension, it is considered a beautiful masterpiece even today.

My world borrows element from Flatland and I need everything in this world to obey Newton's law of universal gravitation, my concern is how can the flatlanders measure gravitational mass?

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    $\begingroup$ Einstein didn't explain gravity any more than Newton did; he merely described it mathematically more accurately then Newton. $\endgroup$ – Mike Scott Sep 9 '16 at 7:03
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    $\begingroup$ What Mike Scott said. Also, for many purposes, Newton's equations are perfectly adequate; I recall reading somewhere that when the US went to the Moon, the error when using Newton's equations (as compared to Einstein's) was on the order of centimeters. I would call that a tolerably good result using several hundred years old math. $\endgroup$ – a CVn Sep 9 '16 at 7:17
  • $\begingroup$ Guys relax my question is focus on applying Newton's law, I'm just giving credit to Einstein who revolutionalise the concept of gravity. As for the force carrier of gravity still remain undetected time will tell. $\endgroup$ – user6760 Sep 9 '16 at 7:23
  • $\begingroup$ Nolo has explained how you can calculate a mass equivalent, but you need to remember that "weight" requires 3 dimensions. $\endgroup$ – Separatrix Sep 9 '16 at 7:41
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    $\begingroup$ Have you looked at physics.stackexchange.com/questions/78975/… ? $\endgroup$ – Cort Ammon Sep 9 '16 at 22:21


When I wrote this answer, having read Flatland many years ago, it escaped my attention that the original story did not have gravity as others pointed out.

My assumption about the question was that to "weigh" a Flatlander, simply "use a scale", but since the original Flatland has no gravity it would seem that no such device would exist. My answer was based on that assumption and that mass is basically given in that way.

As per Salmoncrusher's suggestion one could use a simple, spring based scale with artificial gravity in a rotational frame, using centrifugal force.

You are looking for what is called the contour integral for computing the area of a shape - any shape whose perimeter can be defined as a continuous curve, such as a circle or any parametric curve. For polygons here is a good place to start. It gives an example of a couple of regular polygons and even shows how to compute the area of an irregular polygon. It is important to note that complex polygons, i.e. irregular self-intersecting will not work with the method given there, however your flat landers are not likely to have complex compositions as this would make them very strange indeed.

In two dimensions "mass" is determined by area as it is in 2D physics engines for video games. To keep things simple you could consider one standard unit of area, such as one square meter, to be equal to one standard unit of mass such as a kilogram. However your flatlanders may be composed of different materials with different "densities" which would increase or decrease the unit of mass per unit of area ratio for a higher or lower density shape respectively.

Another point worth noting is that the contour integral for curves can be difficult and sometimes impossible as are some kinds of integrals. Two options are Legendre-Gause Quadrature, which is a more advanced approximation technique, or a conceptually simpler method to a good approximation is to compute many points along the curve, then use the formula for computing the area of an irregular polygon as given above.

  • $\begingroup$ Note to those few who already voted up, and that was fast, but the links are changed to simpler resources so as not to confuse the OP on unrelated subject matter. $\endgroup$ – Nolo Sep 9 '16 at 7:13
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    $\begingroup$ It seems to me that this doesn't actually answer the question. If this is the case, then the flatlanders would never develop independent concepts of area and mass. If densities aren't uniform, how do you measure density? You'll need to measure mass. Back to square one. $\endgroup$ – Salmoncrusher Sep 9 '16 at 18:56
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    $\begingroup$ Oh, I see. I suggest you add the part about the scale in your answer. If there's no gravity in flatland, your scale could operate in artificial gravity from the centrifugal effect. $\endgroup$ – Salmoncrusher Sep 9 '16 at 21:50
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    $\begingroup$ I meant centrifugal. Centripetal force is what keeps objects in circular motion. Centrifugal force is what causes someone standing in a rotating wheel to feel a downwards attraction. Consider a rotating frame of reference where the orbiting body is stationary relative to the coordinate system. The Centripetal force acts inward, the centrifugal force is the equal and opposite force that keeps the system in equilibrium, acting outward. $\endgroup$ – Salmoncrusher Sep 9 '16 at 22:31
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    $\begingroup$ Yes, Centrifugal force is a result of centripetal force as it is the reaction force to centripetal force viewed in a rotating reference frame. I'm not sure how that is related to water swirling down the drain. Are you thinking of the Coriolis effect? $\endgroup$ – Salmoncrusher Sep 9 '16 at 23:02

What type of world are you thinking about? In the story Flatland, the two dimensions are much more similar to a "top down" type of world. In this world, there is no gravitation so there cannot be a gravitational mass. In this world, others have answered how to calculate the inertial mass.

However, if you want universal gravitation, then it might be more appropriate to have the two dimensions resemble a "side view" world, one more like the first Mario game. In this world, you can have one large mass act as a planet, and have processes like locomotion be easier to define than in Flatland.

Two ways to measure gravitational mass in reality are:

  1. A spring scale

Where a mass is placed on a spring (on the ground) and the compression of the spring determines the force applied (the weight of the object). Anywhere the force of gravity is the same, the weight will be the same (excluding charges, non inertial reference frames, etc). Weight will be proportional to mass.

  1. Displacement in a liquid.

Suppose you have a boat floating in a tub, and the tub is filled to the brim with water. If you place an object in the boat (such that the boat is still floating), it will push some water out of the tub because of displacement. The amount of water pushed out weighs exactly the same as the mass in the boat.

I think that both of these methods can easily be adopted in a 2D environment, as long as the environment isn't like the classical "top down" Flatland environment.

  • $\begingroup$ See Plainiverse for that kind of world. $\endgroup$ – JDługosz Sep 12 '16 at 7:35

I can't see a direct way to measure the gravitational mass, but there is a way to measure the inertial mass. In our world, it makes no difference.

Take a standard object. Call it "one kilogram". Throw it at v1 speed on a non-moving target so that the projectile and the target stick after contact (Inelastic Collisions).

Measure the v2 speed after collision (I neglect the ground resistance as there is no ground in 2D).

The inertial mass (in Kilograms) of your object will be:

m = (v1-v2)/v2

enter image description here

Now that you have inertial mass, you have a way to make physics and measure forces. The gravitational force F=m.g is equal to the force you must apply to prevent your object from falling.


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