Continuous Proportions

"Rigidly Defined Areas of Uncertainty"

What do the numbers 36::24::36 mean to you? ...... no doubt vital statistics of a well proportioned figure. Of course we could make the measurements in feet instead of inches giving 3::2::3 and while not being so recognisable the proportions would still be identical. We can in fact change the unit of measurement or we can rescale the sizes (providing the same operation is performed to all the sizes) and the ratios and proportions remain inviolate. If we apply this 3::2::3 pattern to an auricula pip a match can be found with part of Glenny's definition of the ideal auricula.

So strangely enough 36::24::36 also defines the proportions of a Glenny tube!
From the pattern 3::2::3 we can obtain the sizes 2, 3, 5 and 8 from single zones or combining adjacent zones. Coincidentally these numbers are part of the famous Fibonacci series

0, 1, 1, 2, 3, 5, 8, 13, 21, 34 ...... etc

One property of the Fibonacci series is that as the numbers get larger the ratio of adjacent terms gets closer to the Golden Number (1.618) and so the terms are approximately in continuous golden proportion and may be deemed aesthetically good.
If we choose a pattern based on the Golden number of 1.618::1::1.618 the sizes obtained 1, 1.618, 2.618 and 4.236 are in exact continuous proportion and can be considered aesthetically excellent.
The Glenny tube is 25% of the pip and the tube defined as above based on the Golden ratio is a little smaller at 23.6% of the pip.

The proposition is simply that a flower is well proportioned if the zones, singly and in combination, form long sequences of sizes that are in continuous proportion.

There are only four cases where two concentric circles demonstrate good aesthetics by having sizes in continuous proportion.

Case-1 1.618::1::1.618 At 23.6% of the pip, this is of a size suitable for defining the tube. Here all four possible sizes are in continuous proportion. The other instances have just three sizes in continuous proportion.
Case-2 1::1.618::1 At 44.7% of the pip, this could define the boundary between paste and ground.
Case-3 1::2::1 At 50% of the pip, this could also define the boundary between paste and ground. This is a little different as the sizes that are in proportion are 1, 2 and 4 with a common ratio of 2. This obviously agrees with Glenny and Maddock regarding the size of the centre.
Case-4 1::98::1 At 98% of the pip this defines lacing! 98 was chosen because it is a lot larger than the other numbers. Here the sizes in continuous proportion (or nearly so) are 98, 99 and 100 because 99/98 is approximately equal to 100/99, with a common ratio of 1.01 . In a sense the finer the lacing the nearer we get to absolute continuous proportion, but the eye must be able to discern the different sizes and quite robust lacing gives reasonably equal ratios. This could be one reason why cushion plants look better in clay pots.

From the first three cases above we can generate additional concentric circles that have diameters in continuous proportion with existing circles.

Case-1 To a circle at 23.6% a new one at 48.6% can be added defining the boundary between the paste and the ground. (Note that 23.6, 48.6 and 100 are in continuous proportion). Similarly another circle at 69.7% could be added to define the maximum ground. (Note that 48.6, 69.7 and 100 are in continuous proportion).
Case-2 Additional to a circle at 44.7% we could have one at 20% for the tube. (Note that 20, 44.7 and 100 are in continuous proportion). Another circle at 66.9% could be added to define the maximum ground. (Note that 44.7, 66.9 and 100 are in continuous proportion).
Case-3 In addition to a circle at 50% one could be included at 25% for the tube. (Note that 25, 50 and 100 are in continuous proportion). Similarly another circle at 70.7% could be added to define the maximum ground. (Note that 50, 70.7 and 100 are in continuous proportion).

Glenny 25.0 50.0 75.0
Maddock 16.7 50.0 75.0
Case-1 23.6 48.6 69.7
Case-2 20.0 44.7 66.9
Case-3 25.0 50.0 70.7
Numbers are % of pip diameter.

These values for cases exhibiting continuous proportion do not differ radically from the traditional standards and suggest that intermediate tube sizes (between Glenny and Maddock) are quite acceptable as are somewhat smaller centres and grounds. The table below shows the Euclidean Distances (standardised on a distance of 1 from Glenny to Maddock) between the different configurations and implies that Glenny is nearer to the cases of continuous proportion than Maddock.

Euclidean Distances can be used to quantify the difference between patterns by plotting points using coordinates such as tube, paste, ground and edge sizes in multi-dimensional Euclidean space and calculating the distance between the points. As most people, including yours truly, have difficulty visualising distances in four dimensions, here are the answers.
Distances Glenny Maddock Case-1 Case-2 Case-3
Glenny 0 1 0.57 0.85 0.52
Maddock 1 0 1.08 1.07 1.13
Case-1 0.17 0.93 0 0.40 0.15
Case-2 0.62 0.92 0.45 0 0.55
Case-3 0 1 0.17 0.62 0
The cells with a grey background are for Euclidean Distances in three dimensions, with the edge and ground combined into a single zone.

There are of course many other possibilities and the "rigidly defined" florist standards based on tradition, even if there is "uncertainty" over which florist is best, will probably remain sacrosanct. Maybe the answer is to build a mighty computer (as in the Hitchhiker's Guide to the Galaxy) to define what is the perfect standard? So beware "Deep Thought" is under construction and its algorithms are being programmed.

For those more concerned with the simpler selfs, stripes and alpines "Deep Thought" has already suggested a likely candidate. Remember the Fibonacci series, there is a similar sequence (called the Padovan Sequence after the architect Richard Padovan), where the (n)th term is equal to the sum of the (n-3)th and the (n-2)th terms.

0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265 ......

The pattern 49::37::28::37::49 gives the sizes 28, 37, 49, 65, 86, 114, 151 and 200. These are in near continuous proportion with common ratio of 1.325 (the Plastic Number discovered in 1928 by Dom Hans Van Der Laan).
The Plastic Number is the root of the equation x3 - x -1 = 0. 
This obviously post dates Maddock or Glenny. Who knows how the old florists would have defined the perfect auricula given the benefits our current knowledge and technology? A remarkably long neat series, using 8 of the 9 possible sizes, and works only for three concentric circles with their inherent symmetry. There seems to be parallels between the use of the Golden Number for two dimensional geometry and two concentric circles and with the use of the Plastic Number for three dimensional geometry and three concentric circles.
The above gives tube and centre diameter sizes of 14% and 51% of the pip.

Which is far nearer to Maddock's standard than Glenny's for these parts of the pip and not impossible for stripes judging from some of Derek Parsons' latest developments. Most surprising of all is that it is an even closer match (Euclidean distance 0.21) with the defined standards (tube and centre diameter sizes of 15% and 50% of the pip) for another florists' flower, the Gold Laced Polyanthus. If the definition of the size an ideal GLP pip was 21mm instead of 20mm, while keeping the tube size constant at 3mm, an even better match results!
Gold Laced Polyanthus

Before we leave the Padovan Sequence perhaps we should take another look at Maddock.
You can find the values 2, 2, 3, 4, 5, 7, 9 and 12 of the Padovan Sequence below
in the sizes of Maddock's ideal pip above.
0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265 .....

An auricula flower is not quite as simple as just a few concentric circles. There are the effects of feathering of the ground with the edge and the visual impact of the thrum to take into account.

To illustrate this consider Case-3 that has a maximum ground radius of 70.7% of the pip radius with (due to feathering) a maximum leaf-like edge size of 35.3% of the pip radius and a thrum that at least allows the observer to gauge the exact centre of the tube. The patterns below show the tube (size 2) split into a 1::1 pattern as well as a ground of size 0.828 and a leaf-like edge of size 1.414 with a total size of only 2 due to feathering.

2   ::1::  2  ::1::   2
2         ::1::1::1::1::         2
1.414 and 0.828::1::1::1::1::0.828 and 1.414

Can you find the following sizes? 1, 1.414, 2, 2.828, 4, 5.656 and 8.

A reasonably long chain of continuous proportion with a common ratio of 1.414 (the square root of 2) and the sizes used include a lot of single zones and diameters that are relatively easy for the eye to measure. A size of 3 might be substituted for 2.818 and whilst not being so accurate has the advantage of not requiring a perception of the tube centre. I think you would really need a computer program to evaluate how approximations would affect the definition of ideal flower. Perhaps this page should have be entitled "Loosely Defined Areas of Perfection"?


Dr. Ron Knot "Fibonacci Numbers and The Golden Section",
Richard Padovan "Dom Hans Der Laan and the Plastic Number",
Nexus Network Journal, vol 4, no. 3 (Summer 2002),
Ian Stewart "Tales of a Neglected Number", Sci. Amer. 274, 102-103, June 1996.