Montessori knobbed cylinders: the mathematics of radius, height, and volume
Volume of a Cylinder
The volume of a cylinder with radius , height is given by Notice that we have three parameters. If we hold one of them to be constant, then we can obtain the relationship between the other two parameters, especially if one of these two parameters is uniformly or linearly increasing, i.e., a linear function of a set of integers , where is an arbitrary integer ideally greater than or equal to 3.
Let us consider three cases: constant volume, constant radius, and constant height. Each of these cases will have two sub-cases, with each sub-case dictating the design of a row of Montessori knobbed cylinders. Thus, in principle, you can design a total of 6 rows of knobbed cylinders, with each row containing cylinders.
1. Constant volume
If the volume is constant, then the radius of the cylinder is inversely proportional to the square root of the height :
1.1 Linearly increasing height
If we set the height to be an integral multiple of the smallest height , then we may define the height of the cylinder as , where and is the number of the cylinders. The corresponding radius of the cylinder is Notice that the radius is inversely proportional to .
1.2 Linearly increasing radius
If we set the radius to be an integral multiple of the smallest radius , then we may define the radius of the (k^{th}\) cylinder as , where and is the number of the cylinders. The corresponding height of the cylinder is Notice that the height is inversely proportional to .
2. Constant radius
If the radius is constant, then the volume of the cylinder is inversely proportional to the height :
2.1 Linearly increasing height
If we set the height to be an integral multiple of the smallest height , then we may define the height of the cylinder as , where and is the number of the cylinders. The corresponding volume of the cylinder is Notice that the volume is inversely proportional to .
2.2 Linearly increasing volume
If we set the volume to be an integral multiple of the smallest volume , then we may define the volume of the cylinder as , where and is the number of the cylinders. The corresponding height of the cylinder is Notice that the height is proportional to .
3. Constant height
If the height is constant, then the volume of the cylinder is inversely proportional to the square of the radius :
3.1 Linearly increasing radius
If we set the radius to be an integral multiple of the smallest radius , then we may define the radius of the (k^{th}\) cylinder as , where and is the number of the cylinders. The corresponding volume of the cylinder is Notice that the volume is inversely proportional to .
3.2 Linearly increasing volume
If we set the volume to be an integral multiple of the smallest volume , then we may define the volume of the (k^{th}\) cylinder as , where and is the number of the cylinders. The corresponding radius of the cylinder is Notice that the volume is proportional to .
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