## Monday, 19 November 2007

### The Fibonacci number sequence and space bound by hyper-dimensional objects

Volume is defined as "the measure of space taken up by a three-dimensional object". As for example the space bound by a cube which is three-dimensional and is measured in cubic centimetres (cm3) units. But what about the space bound by a four-dimensional object, a tesseract or hypercube. Can we use the same term, volume, to define the space bound? Since the space bound by a tesseract is four-dimensional and not three-dimensional falls out of the boundaries prescribed by the prevailing definition for volume.

In order to get an idea about the space bound by hyper-dimensional objects, we can apply the same simple rules we use to find the space bound by a cube, namely multiply the area times the height of an adjacent side, and for a unit cube you get,

1 cm2 X 1 cm = 1 cm3

the unit for measuring volume, the cubic centimetre (cm3).

By applying the same rule in a tesseract, which has six cube sides, we multiply the volume of one side times the area of an adjacent cube side, so the unit tesseract has

1 cm3 X 1 cm2 = 1 cm5

the quintic centimetre (cm5). The measurement unit of four-dimensional space?

By applying the same rule for a penteract, which has tesseracts as sides, we multiply the space bound by a tesseract (cm5) times the volume of the cube side of an adjacent tesseract (cm3), so the unit penteract has

1 cm5 x 1 cm3 = 1 cm8

So centimetre to the power of 8 (cm8), the measurement unit of five-dimensional space?

Continuing on for a hexeract, which has penteracts as sides, the unit for six-dimensional space is (
cm13).

By looking at the powers of the units of measurement from the known three dimensions to the hyper dimensions, namely length (
cm), surface (cm2), volume (cm3) to four-dimensional (cm5), five-dimensional (cm8) and six-dimensional (cm13), …

the sequence 1, 2, 3, 5, 8, 13 is the Fibonacci numbers sequence.

.. hyper-dimensions ..their evolution ..spiral ..

.. spiral systems .. that the longer they extend ..they become frail ..wither and die ..

.. as they have been observed ..in nature .. (Goethe) ..

.. hyper-dimensions ..have limits ..do not exist beyond these limits .. may ..up to 5 ..penteracts ..

.. and these limits ..can be surmised ..by looking at the limits of ..nature's spiral systems ..

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