Editorial Reviews. About the Author. Author Debbi Miller Gutierrez lives in New Mexico, the The Pinecone Problem by [Debbi Miller Gutierrez]. Kindle App Ad.
Table of contents
- Rabbits, Cows and Bees Family Trees
- Unlocking the Secrets of the Pinecone - Scientific American
- Fibonacci Numbers and Nature
Label the rows A, B and C. For each pinecone, write down the length in the column Initial Length.
Rabbits, Cows and Bees Family Trees
For all measurements in this activity, use centimeters cm. Use your measuring tape to measure the circumference of each pinecone at its widest point. For each pinecone, write down the circumference in the column Initial Circumference. Place pinecone A on the foil-covered baking tray. With the help of an adult, put the tray in the degree F oven.
With the help of an adult, check the pinecone every 10 minutes to make sure it doesn't burn. Are the pinecones changing in any way as they get warmer? What do you notice about them as they get hot? While pinecone A heats up, place pinecone B in the cold water. Use your spoon to hold it underwater. Keep it there for two minutes. What do you notice about the pinecone in the water? Does it sink or float? Why do you think this is true? Do you notice any changes as the pinecone sits under the cold water? Remove the pinecone from the cold water.
Use your measuring tape to measure the length of pinecone B. Write down the length in the column Cold Water Length. Use your measuring tape to again measure the circumference of pinecone B at the widest point. Write down its circumference in the column Cold Water Circumference. Compare the length and circumference of pinecone B in each column. Did its length or circumference change after you put it in cold water? If so, what kind of changes did you notice? Did it get larger or smaller?
Do you notice any other changes about the pinecone? Does it look different? Pine cones are bad for puppies to eat. While they aren't toxic, eating them can cause problems such as intestinal blockages and vomiting. Safer alternatives for chewing exist and care should be taken to avoid pine cones. This usually relates to pine needles or sap, which may be attached to the pine cone your puppy is trying to eat. In addition, anything your puppy eats that isn't formulated for him may make him very sick and lead to vomiting and diarrhea.
The answer to "Are pine cones poisonous to dogs? Both pine cones and pine needles are not poisonous to dogs, however you should not let your dog eat them. The reason for this is because pine needles can perforate the dog's stomach or bowel due to their shape. This is a larger sunflower with 89 and 55 spirals at the edge: Click on each to enlarge it in a new window.
The reason seems to be that this arrangement forms an optimal packing of the seeds so that, no matter how large the seed head, they are uniformly packed at any stage, all the seeds being the same size, no crowding in the centre and not too sparse at the edges. So the number of spirals we see, in either direction, is different for larger flower heads than for small.
On a large flower head, we see more spirals further out than we do near the centre. The numbers of spirals in each direction are almost always neighbouring Fibonacci numbers! Click on these links for some more diagrams of , and seeds. Click on the image on the right for a Quicktime animation of seeds appearing from a single central growing point. The animation shows that, no matter how big the seed head gets, the seeds are always equally spaced. At all stages the Fibonacci Spirals can be seen. The same pattern shown by these dots seeds is followed if the dots then develop into leaves or branches or petals.
Each dot only moves out directly from the central stem in a straight line. This process models what happens in nature when the "growing tip" produces seeds in a spiral fashion. The only active area is the growing tip - the seeds only get bigger once they have appeared. A seed which is i frames "old" still keeps its original angle from the exact centre but will have moved out to a distance which is the square-root of i.
It also has a page of links to more resources. Note that you will not always find the Fibonacci numbers in the number of petals or spirals on seed heads etc.
Why not grow your own sunflower from seed? I was surprised how easy they are to grow when the one pictured above just appeared in a bowl of bulbs on my patio at home in the North of England. Perhaps it got there from a bird-seed mix I put out last year?
Bird-seed mix often has sunflower seeds in it, so you can pick a few out and put them in a pot. Sow them between April and June and keep them warm. Alternatively, there are now a dazzling array of colours and shapes of sunflowers to try.
Unlocking the Secrets of the Pinecone - Scientific American
A good source for your seed is: Nicky's Seeds who supplies the whole range of flower and vegetable seed including sunflower seed in the UK. Have a look at the online catalogue at Nicky's Seeds where there are lots of pictures of each of the flowers. Which plants show Fibonacci spirals on their flowers?
Can you find an example of flowers with 5, 8, 13 or 21 petals? Are there flowers shown with other numbers of petals which are not Fibonacci numbers? Collect some pine cones for yourself and count the spirals in both directions. Soak the cones in water so that they close up to make counting the spirals easier. Are all the cones identical in that the steep spiral the one with most spiral arms goes in the same direction? What about a pineapple? Can you spot the same spiral pattern? How many spirals are there in each direction? Mary's College Maryland USA , Professor Susan Goldstine has a page with really good pine cone pictures showing the actual order of the open "petals" of the cone numbered down the cone.
Fibonacci Statistics in Conifers A Brousseau , The Fibonacci Quarterly vol 7 pages - You will occasionally find pine cones that do not have a Fibonacci number of spirals in one or both directions. Sometimes this is due to deformities produced by disease or pests but sometimes the cones look normal too. This article reports on a study of this question and others in a large collection of Californian pine cones of different kinds. The author also found that there were as many with the steep spiral the one with more arms going to the left as to the right.
On the trail of the California pine , A Brousseau, The Fibonacci Quarterly vol 6 pages pine cones from a large variety of different pine trees in California were examined and all exhibited 5,8 or 13 spirals. Also, many plants show the Fibonacci numbers in the arrangements of the leaves around their stems.
If we look down on a plant, the leaves are often arranged so that leaves above do not hide leaves below. This means that each gets a good share of the sunlight and catches the most rain to channel down to the roots as it runs down the leaf to the stem. Here's a computer-generated image , based on an African violet type of plant, whereas this has lots of leaves. Leaves per turn The Fibonacci numbers occur when counting both the number of times we go around the stem, going from leaf to leaf, as well as counting the leaves we meet until we encounter a leaf directly above the starting one.
If we count in the other direction, we get a different number of turns for the same number of leaves. The number of turns in each direction and the number of leaves met are three consecutive Fibonacci numbers! For example, in the top plant in the picture above, we have 3 clockwise rotations before we meet a leaf directly above the first, passing 5 leaves on the way.
If we go anti-clockwise, we need only 2 turns. Notice that 2, 3 and 5 are consecutive Fibonacci numbers. For the lower plant in the picture, we have 5 clockwise rotations passing 8 leaves, or just 3 rotations in the anti-clockwise direction. This time 3, 5 and 8 are consecutive numbers in the Fibonacci sequence. One estimate is that 90 percent of all plants exhibit this pattern of leaves involving the Fibonacci numbers. These buttons will show the spirals more clearly for you to count lines are drawn between the florets: Each floret is peaked and is an identical but smaller version of the whole thing and this makes the spirals easy to see.
Here are some investigations to discover the Fibonacci numbers for yourself in vegetables and fruit. Take a look at a cauliflower next time you're preparing one: First look at it: Count the number of florets in the spirals on your cauliflower. The number in one direction and in the other will be Fibonacci numbers, as we've seen here. Do you get the same numbers as in the picture? Take a closer look at a single floret break one off near the base of your cauliflower.
It is a mini cauliflower with its own little florets all arranged in spirals around a centre. If you can, count the spirals in both directions. How many are there? Then, when cutting off the florets, try this: Find the next on up the stem. Cut it off in the same way. Repeat, as far as you like and.. Now look at the stem. Where the florets are rather like a pine cone or pineapple. The florets were arranged in spirals up the stem. Counting them again shows the Fibonacci numbers.
Try the same thing for broccoli. Chinese leaves and lettuce are similar but there is no proper stem for the leaves. Instead, carefully take off the leaves, from the outermost first, noticing that they overlap and there is usually only one that is the outermost each time. You should be able to find some Fibonacci number connections. Look for the Fibonacci numbers in fruit. What about a banana? Count how many "flat" surfaces it is made from - is it 3 or perhaps 5?
When you've peeled it, cut it in half as if breaking it in half, not lengthwise and look again. There's a Fibonacci number. What about an apple? Instead of cutting it from the stalk to the opposite end where the flower was , i. Try a Sharon fruit. Where else can you find the Fibonacci numbers in fruit and vegetables?
Why not email me with your results and the best ones will be put on the Web here or linked to your own web page. Why not measure your friends' hands and gather some statistics? When this page was first created back in this was meant as a joke and as something to investigate to show that Phi, a precise ratio of 1. The idea of the lengths of finger parts being in phi ratios was posed in but two later articles investigating this both show this is false.
Fibonacci Numbers and Nature
Although the Fibonacci numbers are mentioned in the title of an article in , it is actually about the golden section ratios of bone lengths in the human hand, showing that in hand x-rays only 1 in 12 could reasonably be supposed to have golden section bone-length ratios. Research by two British doctors in looks at lengths of fingers from their rotation points in almost hands and again fails to find to find phi the actual ratios found were 1: Radiographic assessment of the relative lengths of the bones of the fingers of the human hand by R.
Richard Guy's excellent and readable article on how and why people draw wrong conclusions from inadequate data is well worth looking at: However, the 4 petals of the fuchsia really shows there are plants with petals that are definitely not Fibonacci numbers. Four is particularly unusual as the number of petals in plants, with 3 and 5 definitely being much more common. Here are some more examples of non-Fibonacci numbers: Here is a succulent with a clear arrangement of 4 spirals in one direction and 7 in the other: So it is clear that not all plants show the Fibonacci numbers!