Screen Solutions Lab 1

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Lab 1: Trifurcating Trees

1 Introduction

Bifurcations(2), trifurcations(3), and other multifurcations(4+) are everywhere in na-

ture, when paths split, streams join, trees branch, migration paths diverge, lungs ex-pand, and blood capillaries spread. Here, we examine trifurcating trees, and how mucheach level of branching adds to the total volume of the tree under a specific model ofbranching pattern. Each branch starting from the trunk subdivides into three childbranches a specified number of times, each child branch getting thinner and shorter.We approximate the volume of each branch by a cylinder of fixed radius and length.Recall that the volume of a cylinder is

V = r2l

where r is the radius and l the length. can be entered into mathsheet by simpletyping in pi into any of the cells.The purpose of this lab is also to introduce the spreadsheet and show how it can beused to find simple sums and plot the cumulative summation. The example problembelow covers this aspect.

2 Example Problem

Suppose there are ten sticks labelled 1 through to 10 which measure 10,20,30 throughto 100 cubic centimetres of volume respectively in a linear fashion. Plot as two bar

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graphs: first the set of volumes corresponding to the labels on the sticks, then thecumulative volume of the sticks as their number increases.

3 Solution

Before we plot anything with mathsheet, we first need to input all information there

is to plot. As this is an introduction for many of you to mathsheet, we will proceedthough this example in a step by step fashion. Future labs may contain examples suchas this, but will omit many of the steps here; it is up to you to become familiar with theprogram and how it works in the future. For more information, go to the Mathsheetdocumentation.Text like this you are supposed to type in the spreadsheet, and text like this indicatesa clickable area in MathSheet. In the column labelled a on the spreadsheet, we input

the values of the x coordinates as follows:

Select cell a0 .

Type followed by stick no. and ENTER . The quotation mark tells mathsheetwe are entering a text label, and the label reminds us that we are using this columnfor stick number. By pressing ENTER we set this information into mathsheet,and cause the cell a1 to be selected. Labelling is optional, but will help keep you

organized in future labs.

Now type 1 and ENTER. This will place the number 1.0 in the selected cell a1 .Now the cell a2 is selected.

Type a1 + 1. The spreadsheet interprets this by incrementing the value in a1 bythe value x = 1.

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Select cell a2, grab the little black square at its corner, and drag down to a10 .This will fill the a column with x values at increments of 1 up to x = 10.0. Younow have the x coordinates for your graph, even though you do not need them.

Note: if you click on cells lower in the column, you will see that by copying thecell reference formula in such a way, the cell references themselves are increased

respectively. You will be exploiting this feature of mathsheet often for each lab inthis course.

Hint: For larger data sets, you could click on the cell you with to copy-drag downto select it, then choose Edit then Copy from the top menu. By selecting a numberhere and pressing ENTER mathsheet will copy the desired formula the set numberof times.

We now fill in the desired y values (volume of each stick):

Select cell b0 .

Here we enter text into the cell in order to label the column, so type volume andENTER.

Now that cell b1 is selected, we can start inputting the volumes for each stick one

by one. However, when using tools such as mathsheet, much time can be saved bythinking ahead. Notice first that each volume is conveniently 10 times the sticknumber, so we have a formula we could use.

Sometimes we dont have a formula we can use until after the first number isentered, so we might as well enter the first number by hand to get used to this.Type 10 and ENTER so that value is placed in cell b1.

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Noticing the pattern, we now enter a2*10 into cell b2.

By clicking on b2 to select this cell, then on the little black box in the lowerright hand corner of this cell, hold the click and drag down to b10. This copies ourformula a210 into these cells, modifying the a2 to a3, a4, et cetera as appropriate.

Note: We could also have noticed another pattern, in that each stick volume is10 more than the last. In that case, starting in cell b2 , we could have typed inb1+10 as the first cell entry before copy-dragging down.

Now, it remains to determine the cumulative volume. In cell c0 , we enter the labelcum. vol..

After the first stick, the cumulative volume is the same as the first volume, so

we can type either 10 or b1 , followed by ENTER to enter the information inmathsheet c1 and select cell c2.

For the second value, we can calculate the result and enter 30 into cell c3, or havemathsheet do the work for us and enter in 10+20 or b1+b2.

Note: Mathsheet can be used as a simple calculator whenever you want to calculatea value. Indeed, youll often end up getting better results in this course if you have

mathsheet do your calculations for you whenever possible.

Entering in the values in this fashion, though it will give the correct results, is verytime consuming. You would have to enter numbers for each cell in column c. Inmathsheet, there is most often a better way, by using cell references appropriately.How can you enter the correct value in cell c2 using cell references in such a waythat by clicking on the little black box and dragging down all subsequent values

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are the proper cumulative sums? This is possible, and is left to you as an exercise.If you are having trouble figuring it out, please ask a TA for the course during thelabs, and they will be glad to help.

You should now have a column representing the stick number, a column representingeach sticks volume, and a column representing the cumulative volume of these sticks.

Your spreadsheet should look similar to the table below.

a b c

0 stick no. volume cum. vol.

1 1 10 10

2 2 20 30

3 3 30 604 4 40 100

5 5 50 150

6 6 60 210

7 7 70 280

8 8 80 360

9 9 90 450

10 10 100 550

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4 Plotting Bar Graphs using Mathsheet

We are now prepared to graph this information. Click on the menu Graph and selectopen to open the graph window. In this new window, you will now see the GraphParameters window, where you can change the view region of your data. We want toplot bar graphs of volume and cumulative volume for the sticks, so well use the x axis

to represent stick number and the y axis for volume. Since our sticks range from 1 to10, and volumes from 10 to 550, we set min x to 0, max x to 11, min y to 0, andmax y to 600. If youre not sure of what to enter yet, or wish to perfect your viewwindow, you can always return to this screen by clicking on the Edit menu in thisgraphing window and choosing Graph parameters .

We are now ready to plot the bar chart. To create a new bar graph, click on the Newmenu then choose Bar graph. You will see option to change the colour of the bars,

followed by many other options. In y range you specify the cells in a column whichrepresent the data you which to show. In initial x you specify the x value of the lefthand side of the first bar, and in increment you specify the width of each bar. Strokedraws a black outline around each bar and fill fills each bar with the specified colour.

To generate the graph, mathsheet will plot the first bar in the rectangle between thepoints

(initial x, 0)and

(initial x + increment, value in first cell defined by y range).

The second bar will be drawn between the points

(initial x + increment, 0)

and

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(initial x + 2 increment, value in second cell defined by y range),

and so on for the third and subsequent cells until the value in the last cell defined bythe y range is drawn.

Multiple bar graphs are drawn in the order they are created by New then Bar graph.This means that newly created graphs may obscure older ones. In our case, we want

to plot the information we have in columns b and c. Let us start with c. Since thevalues are higher, if we plotted this graph second it would completely cover the first.

Note: If you find youve created the graphs in the wrong order, do not panic. Simplydelete the first graph by selecting it from the Edit menu (it will be the first bar graphthere), and pressing the delete button. Then, create a new bar graph and re-enter therelevant information.

Since we are graphing the cumulative volumes first, in y range enter the range of cells.To do so, we click on the data entry location to the right of y range and type inc1..c10. In mathsheet, all ranges are given by the first cell followed by two dots thenthe last cell. Since we want our bars to be centred on the whole number (otherwise itwill be hard to tell which number corresponds to which bar), for initial x we enter in0.5, and in interval we enter in 1 . This means that each bar is one unit thick, andthe centre of each bar lies exactly on the corresponding number for each stick. Now

you can see why we chose to use min x to be 0 and max x to be 11 in the graphparameters.

When you are done, it is recommended you colour the graph by dragging two of thecolour sliders all the way to the right (Cyan, Magenta, or Yellow). You can of courseuse your own colours, but this will offset the colour of the second graph (Red, Green orBlue) nicely and print both on a black and white printer in appealing shades of grey.

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Now, create the second bar graph with the same values, except enter in b1..b10 for they range, and colour it either Red, Green, or Blue. Note that if you change the valuesin the mathsheet spreadsheet, the graph will not automatically update. To updateeither graph click on Graph then Open from the spreadsheet, then click on Edit,select the appropriate graph, and press OK .

Label both bar charts by selecting the New menu and choosing the Label option.Enter text in the text box, here Cumulative Volume, then click on OK to register thetext. Now, click on the display where you want your label to be, such as the middle ofthe cumulative volume bar graph. Repeat these steps starting from the New menu foreach label. Proceed now to label the Volume, and dont forget to label each graph youare to hand in with your Name, Student Number, Problem Set Number, and SectionNumber. It is also a good idea to give your graph a title with a larger font point size(such as 16pt). Your end result should look something like the following graph.

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Volume

Cumulative Volume

Sammy Student

12345678

Section 207Problem Set 0

a xamp e

stick number

(cm^3)

(cm^3)

0.0 11.0

0.0

600.0

Figure 1 Example of a plot which is ready to print and hand in. Pleasebe sure to include the appropriate graph labels as well as your name,student number, section number, and problem set number.

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This example was to introduce you to mathsheet. Please do not hand or print thiscompleted work. At this stage, you would be ready to save your work and print. To doso, first log in from the spreadsheet window. Click on File followed by Log in. Enteryour username and password (usually in the format sammystu and S1234567), andenter 103 for the group number. Press OK and you should be logged in. You will nowhave access to all previously saved work by the File and Load menus, and can now

save your work from the File menu by choosing the Save option.

Note: Your account will be deleted at the end of the semester. If you still want touse mathsheet after this time, please contact the lab coordinator.

After you save, you can now print your work. From the graph window, select the Printmenu and choose the to PDF option. This will load the PDF on your computer (andwill exit mathsheet if you use a browser-embedded pdf viewer). From the PDF viewer,

after verifying your work, print the graph.

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5 Problems

5.1 Problem Set 1

A trifurcating tree is formed from the trunk by branching into three new branches,each of which then branches into another three branches and so on. For simplicity,we assume that there are no asymmetries and that each branch is = 0.55 times thelength and = 0.56 times the width of the previous level of branches (or of the trunk).In this way, each level of branches is shorter and thinner than the previous one in thefollowing manner:

ln = ln1; rn = rn1where n is the generation number, ln represents length and rn represents radius suchthat l1 and r1 are the length and radius respectively of the trunk.

There are 8 levels or generations of branches in total, including the trunk, which is3.5m high and has a radius of 0.52m. For this model, assume the trunk and all branchesare cylinders.

Draw a bar graph showing the total volume of each generations branches, wherethe x coordinate is the level of branching (starting at 1). Place the bars such thateach bars centre lies above its number, or more precisely along a grid line indicating

an integer value.

Draw a second bar graph showing the cumulative volume of the branches up togeneration n, i.e. the area of the first bar is equal to the volume of the trunk, the

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area of the second bar is equal to the volume of trunk along with its three offshootbranches, and so on. Plot this graph first to avoid it covering the first graph.

Use the spreadsheet to determine the total volume of the trifurcating tree. Howmuch volume does it fill?

Please hand in one page with both graphs clearly visible and properly labelled (see the

Example in the Instructions). Note that the first bar of the cumulative volume willbe obscured by the trunk volume (as it is the same). Write in the bottom margin thetotal volume of the tree.

Note: This lab is an abstraction. It may not accurately describe the size and scale ofexisting trees and their branches.

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5.1.1 Solution to Problem Set 1

We can organize the spreadsheet as shown in the table below. The radii of the branchesare recorded in the b column, the lengths of the branches in the c column, the volumesin the e column, and the sum of the volumes in the f column. The cumulative volume(sum of volumes) can be done in mathsheet by entering e1 in f1 and e2+f1 in f2,

then copy dragging down from this cell.The graphs are shown in the Figure 2. We find that the total volume is V = 6.13m3

(Notice that your result might be a little different because of rounding errors).

n rn ln 3n1 V(n) V

1 0.52 3.5 1 2.97 2.97

2 0.29 1.93 3 1.54 4.51

3 0.16 1.06 9 0.8 5.31

4 0.09 0.58 27 0.41 5.72

5 0.05 0.32 81 0.21 5.93

6 0.03 0.18 243 0.11 6.04

7 0.02 0.1 729 0.06 6.1

8 0.01 0.05 2187 0.03 6.13

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3

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6

7

0 1 2 3 4 5 6 7 8 9

Figure 2 Volume of each generation un-der the sum of volumes in problem 1

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5.2 Problem Set 2



There are 6 levels or generations of branches in total, including the trunk, which is2.25m high and has a radius of 0.7m. For this model, assume the trunk and all branches

are cylinders.

Draw a bar graph showing the total volume of each generations branches, wherethe x coordinate is the level of branching (starting at 1). Place the bars such thateach bars centre lies above its number, or more precisely along a grid line indicatingan integer value.

Draw a second bar graph showing the cumulative volume of the branches up to

generation n, i.e. the area of the first bar is equal to the volume of the trunk, thearea of the second bar is equal to the volume of trunk along with its three offshootbranches, and so on. Plot this graph first to avoid it covering the first graph.


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Pl h d i ith b th h l l i ibl d l l b ll d ( th I t d ti


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Please hand in one page with both graphs clearly visible and properly labelled (see theExample in the Instructions). Note that the first bar of the cumulative volume willbe obscured by the trunk volume (as it is the same). Write in the bottom margin thetotal volume of the tree.

Note: This lab is an abstraction. It may not accurately describe the size and scale of

existing trees and their branches.

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5 2 1 Solution to Problem Set 2 Int od ction


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We can organize the spreadsheet as shown in the table below. The radii of the branchesare recorded in the b column, the lengths of the branches in the c column, the volumesin the e column, and the sum of the volumes in the f column. The cumulative volume(sum of volumes) can be done in mathsheet by entering e1 in f1 and e2+f1 in f2,then copy dragging down from this cell.The graphs are shown in the Figure 3. We find that the total volume is V = 11.94m3


n rn ln 3n1 V(n) V

1 0.7 2.25 1 3.46 3.46

2 0.36 2.14 3 2.67 6.13

3 0.19 2.03 9 2.06 8.19

4 0.1 1.93 27 1.59 9.77

5 0.05 1.83 81 1.22 11

6 0.03 1.74 243 0.94 11.94

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5 3 Problem Set 3 Introduction


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5.3 Problem Set 3



There are 7 levels or generations of branches in total, including the trunk, which is3.9m high and has a radius of 0.25m. For this model, assume the trunk and all branches

are cylinders. Draw a bar graph showing the total volume of each generations branches, where

the x coordinate is the level of branching (starting at 1). Place the bars such thateach bars centre lies above its number, or more precisely along a grid line indicatingan integer value.




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Please hand in one page with both graphs clearly visible and properly labelled (see theExample in the Instructions). Note that the first bar of the cumulative volume willbe obscured by the trunk volume (as it is the same). Write in the bottom margin thetotal volume of the tree.


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5.3.1 Solution to Problem Set 3 Introduction


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n rn ln 3n1 V(n) V

1 0.25 3.9 1 0.77 0.77

2 0.15 2.15 3 0.45 1.22

3 0.09 1.18 9 0.27 1.494 0.05 0.65 27 0.16 1.65

5 0.03 0.36 81 0.1 1.75

6 0.02 0.2 243 0.06 1.8

7 0.01 0.11 729 0.03 1.84

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2 Introduction


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5.4 Problem Set 4 Introduction


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A trifurcating tree is formed from the trunk by branching into three new branches,each of which then branches into another three branches and so on. For simplicity, weassume that there are no asymmetries and that each branch is = 1 times the lengthand = 0.55 times the width of the previous level of branches (or of the trunk). Inthis way, each level of branches is shorter and thinner than the previous one in thefollowing manner:







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n rn ln 3n1 V(n) V

1 0.68 2.7 1 3.92 3.92

2 0.37 2.7 3 3.56 7.48

3 0.21 2.7 9 3.23 10.714 0.11 2.7 27 2.93 13.64

5 0.06 2.7 81 2.66 16.3

6 0.03 2.7 243 2.41 18.72

7 0.02 2.7 729 2.19 20.91

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5.5 Problem Set 5 Introduction


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A trifurcating tree is formed from the trunk by branching into three new branches,each of which then branches into another three branches and so on. For simplicity,we assume that there are no asymmetries and that each branch is = 0.65 times thelength and = 0.66 times the width of the previous level of branches (or of the trunk).In this way, each level of branches is shorter and thinner than the previous one in the

following manner:ln = ln1; rn = rn1

where n is the generation number, ln represents length and rn represents radius suchthat l1 and r1 are the length and radius respectively of the trunk.






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Please hand in one page with both graphs clearly visible and properly labelled (see theE l i th I t ti ) N t th t th fi t b f th l ti l ill

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We can organize the spreadsheet as shown in the table below. The radii of the branchesare recorded in the b column, the lengths of the branches in the c column, the volumesin the e column, and the sum of the volumes in the f column. The cumulative volume(sum of volumes) can be done in mathsheet by entering e1 in f1 and e2+f1 in f2,then copy dragging down from this cell.

The graphs are shown in the Figure 6. We find that the total volume is V = 19.74m3


n rn ln 3n1 V(n) V

1 0.64 3.7 1 4.76 4.76

2 0.42 2.41 3 4.04 8.81

3 0.28 1.56 9 3.44 12.244 0.18 1.02 27 2.92 15.16

5 0.12 0.66 81 2.48 17.64

6 0.08 0.43 243 2.11 19.74

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