Mathematical Recreations : Ramanujan's Nested Radical

Srinivasa Ramanujan posed the following puzzle to the Journal of Indian Mathematical Society in April 1911.

What is the value of the following?

\( \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{\cdots}}}} \)

He waited over six months, and when nobody replied he gave his solution. The result he provided was 3.

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The algebraic solution provided by Ramunujan:

Consider the identity 

\( (x+n)^2 = n^2 + 2nx + x^2 = n^2 + x[(x+n)+n] \)

Take square roots to get

\( [x+n] = \sqrt{n^2+x[(x+n)+n]} \)

Now inside the brackets you have (something + n), so you can
substitute in the same equation with (x+n) replacing (x) to get

\( x+n = \sqrt{n^2+x\sqrt{n^2+(x+n)[(x+2n)+n]}} \)

and repeat the process to get

\( x+n = \sqrt{n^2 + x\sqrt{n^2+(x+n)\sqrt{n^2+(x+2n)\sqrt{\cdots}}}} \)

If you now set n = 1, x = 2 you get

\( 3 = \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{\cdots}}}} \)

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An alternate way of calculating the equation reveals some interesting behavior.

\( 3 = \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{\cdots}}}} \)

\( 3^2 = 1+2\sqrt{1+3\sqrt{1+4\sqrt{\cdots}}} \)

\( 3^2-1= 2\sqrt{1+3\sqrt{1+4\sqrt{\cdots}}} \)

\( \frac{3^2-1}{2}= \sqrt{1+3\sqrt{1+4\sqrt{\cdots}}} \)

\( 4 = \sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{\cdots}}}} \)

\( \vdots \)

\( n = \sqrt{1+(n-1)\sqrt{1+(n)\sqrt{1+(n+1)\sqrt{\cdots}}}} \)

The value of each sequential nested radical forms a series...

\( 3, 4, 5, 6, \cdots, \infty \)

...which is perfectly valid. If 3 was not a valid answer to the equation, then this calculation would have revealed a contradiction. Proof by contradiction is one of my favorite methods of proving.

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So, what happens if we start with a different starting assumption?

\( 4 = \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{\cdots}}}} \)

\( \frac{3\cdot5}{2} = \sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{\cdots}}}} \)

\( \frac{13\cdot17}{2^{2}3} = \sqrt{1+4\sqrt{1+5\sqrt{1+6\sqrt{\cdots}}}} \)

\( \frac{11\cdot19\cdot223}{2^{6}3^{2}} = \sqrt{1+5\sqrt{1+6\sqrt{1+7\sqrt{\cdots}}}} \)
\( \frac{17\cdot27902701}{2^{12}3^{4}} = \sqrt{1+6\sqrt{1+7\sqrt{1+8\sqrt{\cdots}}}} \) 

\( \vdots \)

The value of each sequential radical forms another series.

\( 4, 7.5, 18.41\bar6, 84.543402\bar7, 1429.71739065, \cdots, \infty \)

It is harder to predict subsequent entries of the series than for the original series, but no less valid. I've put some effort into constructing an algebraic function that also produces this series, but have set aside the challenge for now.

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So, let's try another starting assumption.

\( 2 = \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{\cdots}}}} \)

The value of each sequential radical forms a third series...

\( 2, 1.5, 0.41\bar6, -0.206597\bar2, -0.191463517554, \cdots, -\infty \)

...which is not valid under the convention that a radical can't have a negative value.

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Generalizing this computation method, Ramanujan's Nested Radical has an infinite number of valid answers...

\( 3 \ge \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{\cdots}}}} \)

Every starting value greater than 3 results in a series of radical values that increase at more than 1 per step and increase to positive infinity. Every starting value less than 3 results in a series of radical values that increases at less than 1 per step and eventually turns negative. I don't yet know how to prove this general statement, but the pattern holds. The system behaves very simply when the starting value is 3, more than 3, or less than 3.

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Ramanujan's solution to his puzzle is correct, but it is also incomplete.

The algebraic method is unable to find the full solution to the equation. Algebra doesn't effectively deal with infinites, so it shouldn't entirely be a surprise that an algebraic intuition might make some faulty (or at least incomplete) inferences when infinity is involved. This is interesting because it hints at the limits of algebra as a way of knowing truth. The Gödel Incompleteness Theorems discuss something similar to this and are worth reading more about.

How does this connect with biology?

One detail of my training to be a scientist is learning to look for untested assumptions in arguments that are presented to me. This was never explicitly stated in any of my graduate coursework, but it has been important to do well in the academic environment I experienced in getting my PhD. When I find such an assumption, I see it as a puzzle to be explored. Am I seeing something real? What inferences is the presenter drawing from the assumption? Why is what I'm seeing not already addressed?

What assumptions have you made today?

References:

1. Ramanujan: www.thefamouspeople.com/profiles/srinivasa-ramanujan-503.php
2. Algebraic solve: mathforum.org/library/drmath/view/52674.html
3. Numerical solve: www.johndcook.com/blog/2013/09/13/ramanujans-nested-radical/ 
4. Proof by contradiction: en.wikipedia.org/wiki/Proof_by_contradiction
5. Incompleteness theorems: en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

Mystery Plant ID Found!

1. omnilexica.com/?q=silphium

A little over a year ago, I was involved in a wedding party and we were walking through a park to a site for some of the arranged photographs. Along the walk, we crossed a stream and there I came across an interesting plant.

The plant was done flowering, but the seed heads retained a nice green color and had the lovely geometry seen in image #1. I picked a sample of the plant, using the pocket-knife I almost always carry (largely for such purposes as this).

Neither myself, my mother, nor my aunt were able to identify the plant. Now this was getting interesting, as my mother and aunt have a long history of gardening and interest in wildflowers. My aunt noted the material had the scent of sunflowers, but this only helped identify it to the family level, which the structure itself had already provided. I tucked the sample into my pocket, hoping to keep it intact until I could identify it at some later time. On returning home, I consulted the field guides I owned. I had no luck, since they all seem to focus on the most charismatic aspect of the wildflowers…  usually, the flowers themselves.

As the plant material dried, I realized that I had collected very mature looking seeds of this unknown plant. I cleaned the seeds and stored them in a vial, hoping to identify them when I later had the chance to grow plants from them. Part of the motivation was also because I thought that the green seed heads would make a nice display in a vase, so I could use the plant as part of my home flower cutting garden.

2. diggingdog.com/pages2/plantpages.php/P-1403

…a year passes…

I am in the final stages of completing my PhD thesis and found my mind wandering. I was reading about the efforts to breed perennial versions of annual crops and to domesticate new perennial crops from wild plants. While looking through a document on the efforts to generate perennial seed crops, I saw a dried plant specimen of Silphium integrifolium (Deam's Rosinweed) and it struck a memory. A quick web-search later and I found an image of exactly what I discovered on the creekside. However, there are several potential species in the genus Silphium that live in the area and have the memorable seed heads, so it will still take growing the seeds I saved to identify the plant down to the species level.

The plant was being studied for its domestication potential as a new seed crop. I still think it would be a worthwhile project to domesticate it as green material for florist use. Domestication of a single plant species can easily go in multiple directions at once, as the plethora of cabbage/broccoli/etc. plants indicate.

Now that I know what the plant looks like and that it will produce some really nice flowers (image #2), I can better plan for where the plant will go in my garden next year.

References:

• www.csu.edu.au/research/grahamcentre/conferences_workshops/perennial_grains/pdfs/14%20David%20Van%20Tassel.pdf
• seed head 1: www.omnilexica.com/?q=silphium
• seed head 2: luirig.altervista.org/flora/taxa/floraspecie.php?genere=Silphium
• flower: diggingdog.com/pages2/plantpages.php/P-1403

GMO labeling : The right to know what is in our food.

Sometimes biology intersects with political-charged topics that engage large-numbers of people. In the USA, there has been recent activity around the labeling of foods made from genetically modified organisms (GMOs). The following video epitomizes one viewpoint on the subject.

http://www.upworthy.com/a-14-year-old-explains-food-labeling-in-language-even-condescending-tv-hosts-should-get-3?c=reccon3

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One of the major arguments presented in the video is that people have the right to know what is in our food. Because of this, GMO foods should be labeled.

I think this is a wonderful idea!

However… there's a big problem. People think they know what is in regular crops, while they think they don't know what is in GMO crops. Can you list the compounds found in the last organic heirloom tomato that you ate? Do you know what is in "natural" corn? How about the poisons found in the various types of beans that people eat? How about the poisons sprayed on any crops (even organic) that you might find in the store? The reality is that people, in general, have no idea what is in any of the food they eat.

A second problem I have is that the GMO-labeling ideas being pushed will do nothing to assist people in knowing what is in their food. If you're eating a tomato which included a gene from a fish and the label only says the tomato has been altered with a certain form of technology, then the label does nothing at all to inform you about what you're eating.

We do already have an established method for indicating the presence of small amounts of diverse substances in our food that can be utilized for labeling GMOs: the ingredients list. If your tomato product also includes some fish genes, then the ingredients list for the tomato should include the fish (and any bacterial) genes that are found in it. Finding an ingredients label on a single tomato will discourage those who aren't interested in purchasing GMOs, but will allow those who are interested in learning what is in their food to be able to easily educate themselves.

So far, no organization is pushing for such an educational labeling idea. The people pushing for GMO-labeling don't want honest and educational labeling. They're pushing for labels to inspire fear, uncertainty, and doubt (FUD) about GMOs.

I think such a labeling system should also be extended to indicate what chemicals were sprayed on the fruit and thus are likely to remain on the fruit. This issue doesn't involve GMOs, but these chemicals are something we do know with certainty are poisons...  and yet there is no push for this information on labels.

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I am a biologist, so I realize my perceptions of this subject are likely to be distinct from those of most people.

I agree with the reasons stated for GMO labeling, but I am totally against every GMO-labeling proposal I have so far heard being discussed because they don't help attain the goal of the stated reasons.

Hybrid sunflower roots.

1. Plant 3, roots & tuber.

A few weeks ago, we had a solid freeze and the sunflower season came to an end. I dropped by the old place and dug up my sunflowers to see if any of the F1s had developed tubers. The largest hybrid (plant 3) produced one skinny tuber, while the smaller two plants appeared to produce no tubers at all. This was a disappointment, but a more detailed examination of the remaining plant material led to some positive surprises.

2. Plant 1, seeds.

The first plant looked very much like the Helianthus tuberosus mother, but with more red pigment on the stems. I had assumed this was the result of some recombination of maternal alleles and was at best a control to compare the hybrid plants against. When I looked at the remaining two dried flowers on this plant and found them to be full of seeds (image #2), I realized this plant is likely to also be a hybrid.

3. Plant 3, root bud.

The plant that grew the tuber in image #1 also produced a second type of perennial structure, a new bud growing from the old root crown (image #3). This is a structure that isn't seen in either parent species. This feature has been observed before in crosses of this type, however, so I should have expected the possibility.

4. Plant 2, roots.

The other two plants had mildly-swollen roots (images #4 & #5) that have appeared alive when I checked on them. I don't know if they will show new growth in the spring, or if the roots will at some point finally start to rot.

5. Plant 1, roots.

I'm storing the tuber and all the root structures in a dedicated small cube-fridge over the winter. This should keep them safe from the mice that wander through my basement and allow me to grow at least one of them again next year alongside additional F1s from the ~70 seeds that remain from the first cross.

References:

1. H. annuss x H. tuberosus : bulbnrose.x10.mx/Heredity/sunflowerXchoke/sunflowerXchoke.html

Black Nightshade

I've recently written a new post on the topic of toxicity in this plant that you should also read if you've found this page via search or other methods.

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1. Solanum nigrum.

Myths of edibility are much shorter-lived than myths of toxicity. If something is poisonous and you keep eating it, you (or your surviving friends and relatives) will soon learn your error. If something is perfectly edible, but you never eat it for fear of poison, then you will never learn what you're missing.

The common weedy plant Solanum nigrum (Black Nightshade) is a premiere example of this. The berries are routinely considered to be poison, even though there are no recorded fatal poisonings unambiguously associated with the plant.

The berries of every S. nigrum plant I've come across have been very edible, tasting like a somewhat floral and mildly sweet tomato. The ripe berries and green leaves are used over much of the world, with the leaves being used as a pot-herb comparable to spinach. Research has suggested that it might not be a good idea to eat the unripe berries, as they sometimes contain a limited amount of solanine.

Plant poisons tend to be polite, in that they have the trait of tasting poisonous. The putative toxin in S. nigrum is solanine, which has a bitter taste. There are reports of some S. nigrum plants having bitter leaves and unpleasant berries, while others have bland leaves and mildly sweet berries. I suppose I should advise you to not eat the unpleasant tasting plants.

The myth of toxicity of S. nigrum seems to have been spread by the European diaspora. European-derived cultures everywhere seem to think it is deadly poison, even while the natives living in the same places continue eating it routinely. Why would Europeans think this plant is poison?

2. Atropa belladonna; UK range map.

In the UK and much of western Europe, there grows another plant with black berries. This one, Atropa belladonna (Deadly Nightshade), is deadly poisonous, with a long recorded history of deaths… but only in Europe where it grows. In Europe, if you taught children that one black berry (S. nigrum) was edible, but another (A. belladonna) was poison, there would be the risk of them making a deadly mistake in identification. In this context, it is perfectly reasonable for European parents to teach their children that black berries are poison.

A. belladonna has spread to a few other places in the world, but isn't something you will generally run into. If you don't know plants well enough to tell the difference between S. nigrum and A. belladonna, then you really shouldn't be eating anything you find outside. The plants are as distinct as a dog is from a cat. You almost assuredly have experience with identifying those animals, so there is absolutely no way you would mistake one for the other. It still is a good idea to teach children not to eat things you can't identify, but you shouldn't be claiming poison is the reason.

3. Diospyros texana.

The aversion to black berries has even carried over to entirely unrelated plants, that just happen to have black, round fruit.

Diospyros texana (Texas Persimmon) is a tree that produces perfectly edible black fruit that many (of European cultural extraction) consider to be poisonous, even though there are no toxic relatives or mimics. It has a long history of utilization as a food source by American natives of the arid Southwest, but has in recent times been marginalized to a landscaping plant because of the peculiar attitudes of the now-dominant culture.

4. S. nigrum.

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Forms of S. nigrum have been partly domesticated under the name "Garden Huckleberry". These plants have slightly larger berries and a more upright growth form than most of the wild plants. There are red ("Makoi") and orange ("Otricoli") varieties that people might be more likely to believe are edible.

I've collected numerous seeds from a local (Minnesota) form of S. nigrum, with the goal of using them in a mutation breeding experiment. The basic idea is to expose a batch of seeds to some mutagen, like X-rays or some chemicals, and then grow out the resulting plants to look for variations which might be more useful. Larger or different colored fruit are the most obvious things to look for, but other interesting traits may also appear. I would like to use ultraviolet light as a source of mutations, as UV-light is easy to control and keep contained, but I still need to determine if it will work for these seeds.

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There are still occasional reports of people eating S. nigrum and experiencing gastrointestinal distress. They could have had a specific allergic reaction to the new food source. For this reason, people should be conservative about eating plants they don't have experience with.

5. Solanine-rich S. dulcamara.

The putative poison found in S. nigrum is the bitter-tasting solanine. It is not entirely clear if everyone can taste this compound. You can experimentally determine your ability to taste Solanine by tasting the very common S. dulcamara (image #5), which has elongated orange/red berries and purple flowers. S. dulcamara is definitely toxic due to the high levels of solanine found in its leaves and berries. For several years, I have been occasionally tasting the berries (looking for a 'sweet' version), but have found very little variation in the amount of poison. If the fruit of this plant tastes sweet to you, then you should have someone else taste it before you really eat any and you might want to avoid tasting wild things like this as a rule.

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If you eat some S. nigrum (or S. dulcamara) berries and get sick, you really can't blame me for it. "Some guy on the internet told me it was ok!" won't hold up in court.

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References:

1. Solanum nigrum
• en.wikipedia.org/wiki/Solanum_nigrum
• www.liseed.org/solanunusual.html
• black: www.rareseeds.com/garden-huckleberry/
• red: www.flickr.com/photos/sugans_shutters/10223815933/
• orange: www.rareseeds.com/otricoli-orange-berry/
2. Atropa belladonna
• en.wikipedia.org/wiki/Atropa_belladonna 
• www.lmarrick.com/ancient-anesthetics-belladonna/
• www.brc.ac.uk/plantatlas/index.php?q=plant/atropa-belladonna
3. Diospyros texana
• en.wikipedia.org/wiki/Diospyros_texana
• www.texasbeyondhistory.net/st-plains/nature/images/persimmon.html
4. Solanum dulcamara
• en.wikipedia.org/wiki/Solanum_dulcamara

Nicotine

Nicotine is highly addictive, and not just for us humans.

I observed this black vulture (Coragyps atratus) apparently looking for and eating discarded cigarettes at a roadside park in central Florida. I also noted the same behavior with a different group of vultures at another park during the same trip.

If we were to eat tobacco like this, it would quickly make us sick. Vultures are adapted to eat decaying meat and all the nasty toxins that go with it. Because of this, they have what would be described as an, "iron stomach". I wonder how common the behavior is and if addicted vultures get angry when they haven't had a fix in too long.

Cigarette butts have reportedly been found in the intestines of whales (presumably having been washed out to the ocean and ingested along with their normal food). There are photos of other birds investigating cigarette butts they come across, but I haven't found any reports of any animals actually eating them.

(While trying to find other reports of this behavior, I did find a report of people smoking dried vulture brains. People are strange things.)