Aliens: an informative response

Background: The SETI endeavour is based upon the assumption that outer space is pervaded by life forms of one kind or another; and that included among these must be some that possess an intelligence equal or superior to our own. It is further held that one or more of these alien civilisations, even now, are actively engaged in advertising their presence and capabilities by broadcasting carefully designed electromagnetic signals to all corners of the cosmos. We here assume that such a signal has been received (typically, a long repeated sequence of large prime numbers), and consider how we might formulate a positive response that would contribute to the longer term objectives.

Consider the following:

• The first billion (1,000,000,000) natural numbers include the following set of 8 triangular numbers which are distinguished by the fact that their sides and perimeters are also triangular:

{6, 21, 666, 3081, 123256, 584821, 23829156, 113273826}

• Here is a demonstration of these characteristics in respect of the third, 666:

• The number of units forming a side is clearly 36 (or 6x6); and the perimeter, or outline has 3x35, or 105. These triangles are depicted in Figure 2 along with another absolute triangular feature which is descriptive of all triangles.

• The following table reveals how the number we refer to as 'six-hundred and sixty-six' would appear as a group of symbols in various positional number systems, radices 2 - 36, inclusive.

[Observe that the standard set of denary digits, 0-9, is here augmented by the upper case alphabetics, A-Z]

 

While there are a number of patterns of visual interest here (eg the complementary reflection in the 8 digits of the binary representation, and the palindromic '22122' associated with radix 4), the one which is most likely to catch the attention of our intelligent alien is '666' because the eye-catching digit repetition involves the first perfect number, and provides further examples of triangularity - as revealed in Figure 3. In addition, 10 (the corresponding radix) is itself triangular.

But there are further reasons for our alien interpreter to seize upon the information that earthlings are essentially denary beings - basing their representation of numbers and physical measurements on ten as collective unit. For example, the sum of the absolute features of Figure 2 is 144 - a perfect square; the sum of the radix-dependent features of Figure 3 is precisely half that, viz 72; and the total of all these triangular features of 666 is 216, or 6x6x6!

Observe then that the unique cube (surface area numerically equal to volume) appears as a result of this analysis.

To summarise our proposed response:

• Transmit the sequence {6, 21, 666, 3081, 123256, 584821, 23829156, 113273826} repetitively to capture the attention of our alien

• Intersperse transmission with 666 repeated 666 times - thus drawing attention to this particular number

• The intelligence of the receptor should accomplish the rest

Clearly, the above would represent just one segment of an informative response - but, I suggest, an important one: the fact that 666 would be recognised as the first triangular multiple of the uniquely symmetrical number 37 should provide a solid basis for the transmission of digitised images.

Vernon Jenkins MSc

2002-12-4