Critical mass is the minimum amount of a substance that is needed to sustain a reaction. This amount can vary based on how it is shape and the density of the substance. When a reaction occurs it emits neutrons that are capable of producing more reactions but only if they collide with more atoms. Higher probability of a collision means more collisions which is a more efficient reaction.

When a reaction starts an atom is split and neutrons are emitted. Optimally, all neutrons hit other atoms because if they do not then the overall reaction is less efficient because time emphasizes these errors as shown in the second table. If an atom is completely surrounded by other atoms when it gets split then its neutrons have a 100% chance of hitting more atoms. A 100% chance means no waste, at least for that particular atom. If an atom is on the surface of the mass then its neutrons have a chance to go away from the mass and not produce another reaction. If it is within the mass they are forced to react with more atoms because no matter what direction they go in they cannot avoid splitting an atom. This means that we want to minimize the mass's surface area and maximize its volume which is controlled by the shape of the mass.

Some shapes have excellent surface area to volume ratios. Here are calculations for the surface area of various shapes given 1000 volume:

Plane: 2000
Cube: 600
Rectangle: 700
Sphere: ~9

As you can see a sphere is by far the best, only about 1% of the volume is exposed to the surface. This is because spheres only have one surface and no corners. Also consider that a sphere cannot be tangent to a plane at more than one point, planes being the worst for SA:V ratios. Spheres produce better reactions because they have the smallest surface area to volume ratio.

The last factor that controls critical mass is density. When the density of the mass is increased, the neutrons cannot go as far without colliding so it makes the chain reaction faster. Critical mass is just enough mass to make the reaction continue happening. If the best shape and highest density is used then the quantity of mass required for a sustained reaction or the critical mass is lessened.

Here is a table illustrating how fast the total reaction is going with different quantities of successful (colliding) neutrons:

```    2    3    4
-------------------
1|  2    3    4
2|  4    9   16
3|  8   27   64
4| 16   81  256
5| 32  243 1024
```

The smaller the number of successful reactions the more dramatic the results are later on. In practice since it is determined by probability whether or not a collision fails to occur it is not this bad unless the probability is very low. The chain increases in total reactions exponentially over time so the longer it has been going the worse the effect of a miss.

In summary, critical mass is the quantity of a substance needed to sustain a chain reaction and it is directly dependent upon the shape and density, and when these factors are manipulated correctly (spheres and high density) critical mass is lessened.