Research work

The fractal structure’s researching.

Modeling of the fractal sets in the Maple program.

Municipal Budget Educational Establishment “School # 57” of Kirov district of Kazan

    Author:  Ibragimova Evelina

    Scientific advisor:  Alsu Gibadullina - mathematics teacher

    Translator:  Aigul Gibadullina

In Russian

ИбрагимоваЭ_Фракталы.docx

In English

Fractals_researching.doc

     ( Images - in attached files )

Table of contents:

Introduction

I. Studying of principles of fractals construction

II. Applied meaning of fractals

III. Researching of computer programs of fractals construction

Conclusion

Introduction

We don’t usually think about main point of things, which we have to do with every day. Environmental systems are many-sided, ever–changing and compli­cated, but they are formed by a little number of rules. Fractals are apt example of this – they are complicated, but based on simple regulations. Self – similarity is the main attribute of them.  Just one fractal element contains genetically information about all system.  This information have a forming role for all system. But some­times self – similarity is partial.

Hypothesis of the research. Fractals and various elements of the Universe have general principles of structural organization. It is a reason why the theory of frac­tals is instrument for cognition of the world.

Purpose of the research. Studying  of genetic analogy  between fractals and alive and non-living Universe systems with computer-based mathematical mod­eling in the Mapel’s computer space.

Problems of the research. 

1. studying of principles of fractal’s construction;
2. Detection of  general fractal content of physical, biological and artificial sys­tems;
3. Researching of applied meaning of fractals;
4. Searching of computer programs which can generate all of known fractals;
5. Researching of fractals witch was assigned by complex variables;
6. Formation of innovative ideas of using of fractals in different spaces;

The object of research.  Fractal structures, nature and society objects.

The subject of research. Manifestation of fractality in different objects of the Universe.

Methods of the researching.

1. Studying and analysis of literature of research’s problem;
2. Searching of computer programs which can generate fractals and experimenta­tion with them;
3. Comparative analysis of principles of generating of fractals and structural or­ganizations of physical, biological and artificial systems;
4. Generation and formulation of innovation ways to applied significance of fractals.

Applied significance.

Researching of universality of fractals gives general academic way of cognition of nature and society.

I. Studying of principles of fractal construction

We can see fractal constructions everywhere – from crystals and different accu­mulations (clouds, rivers, mountains, stars etc.) to complex ecosystems and bio­logical objects like fern leafs or human brain. Actually, the idea that frac­tal principles are genetic code of our Universe has been discussed for about fifteen years. The first attempt of modeling of the process of the Universe construction was done by A.D. Linde. We also know that young Andrey Saharov had solved “fractal” calculation problem – it was already half a century ago.

Now therefore, fractal picture of the world was intuitively anticipated by human genius and it inevitably manifested in its activity.

Fractals are divided into four groups in the traditional way: geometric (constructive), algebraical (dynamical), stochastical and natural.

The first group of fractals is geometric. It is the most demonstrative type of fractals, because we can instantly observe the self-similarity in it. This type of fractals is constructed in the basis of original figure by her fragmentation and real­izing of different transformations. Geometrical fractals ensue on repeating of this procedure. They are using in computer-generated graphics for generating the pic­tures of leafs, bush, dimensional structures, etc.

The second large group of fractals -  algebraical. This fractals are constructed by iteration of nonlinear displays, which set by simple formulas. There are two types of algebraical fractals – linear and nonlinear. The first of them are determined by first order equates (linear equates), and the second by nonlinear equates, their na­ture significantly brighter, richer and more diverse than first order equates.

The third known group of fractals – stochastical. It is generated by method of random modification of options in iterative process. Therefore, we get an objects which is similar to nature fractals – asymmetrical trees, rugged coasts, mountain scenery etc. Such fractals are useful in modeling of land topography, sea–surface and electrolysis process etc.

The fourth group of fractals is nature, they are dominate in our life. The main difference of such fractals is that they can’t demonstrate infinite self-similarity. There is “physical fractals” term in the classification concept for nature fractals, this term notes their naturalness. These fractals are created with two simple opera­tions: copy and scaling. We can indefinitely list examples of nature fractals: hu­man’s circulatory system, crowns and leafs of trees, lungs, etc.  It is impossible to show all diversity of nature fractals.

II. Applied meaning of fractals

Fractals are having incredibly widespread application nowadays.

In the medicine. Human’s organism is consists from fractal structures: circulatory system, bronchus, muscle, neuron system, etc. So it’s naturally that fractal algorithms are useful in the medicine. For example, assessment of rhythm of fractal dimension while electric diagrams analyzing allows to make more infor­mative and accurate view on the beginning of specific illnesses. Also fractals are using for high–quality processing of  X–ray images (in the experimental way). There are designing of new methods in the gastroenterology which allows to ex­plore gastrointestinal tract organs qualitative and painlessly. Actually, there are discoveries of application of fractal methods for diagnosis and treatment of cancer.

In the science. There are no scientific and technical areas without fractal calcu­lations nowadays. It happens due to the fact that majority of nature objects have fractal structures and dimension: coasts of the continents; natural resources alloca­tion; magnetic field anomaly; dissemination of surges and vibrations in an elastic environments; porous, solid and fungal bodies; crystals; turbulence; dynamic of complicated systems in general, etc. Fractals are useful in geology, geophysics, in the oil sciences… It’s impossible to list all the spaces of adaptability.

Modeling of chaotic processes, particularly, in description of population models.

In telecommunications. It’s naturally that fractals are popular in this area too. Natan Coen is person, who had started to use fractal antennas. Fractal antenna has very compact form which provides high productivity. Due to this, such antennas are used in marine and air transport, in personal devises. The theory of fractal an­tennas has become an independent, well-developed apparatus of synthesis and analysis of electric small antenna (ESA) nowadays. There are developments of possibility of fractal compression of the traffic which is transmitted through the web. The goal of this is more effective transfer of information.

In the visual effects. The theory of fractals has penetrated area of formation of different kinds of visualizations and creation of special effects in the computer graphics soon. This theory are very useful in modeling of nature landscapes in computer games. The film industry also has not been without fractal geometry. All the special effects are based in fractal structure: mountain landscape, lava, flame, fog, large flows and the same. In the modern level of the cinema creation of the special effects is impossible without modeling of fractals.

In the economics. The Veirshtrass’s function is famous example of stochastic fractals. Analysis of graph of the function in interactive mathematic environment Maple allows to make sure in fractal structure of function by way of entry of dif­ferent ranges of graphic visualization. In any indefinitely small area of the part graph of the function absolutely looks like area of this part in the all . The property of function is used in analysis of stock markets.

In the architect. Notably, fractal structures have become useful in the architect more earlier than B. Mandelbrode had discovered them. S.B. Pomorov, Doctor of Architecture, Professor, member of Russian Architect Union, talks about applica­tion of fractal theory in the architect in his article. Let’s see on the part of this arti­cle:

“Fractal structures were found in configuration of African tribal villages, in an­cient Vavilon’s ziggurats, in iconic buildings of ancient India and China, in gothic temples of ancient Russia .

We can see the high fractal level in Malevich’s Architectons. But they were cre­ated long before emergence of the notion of fractals in the architect. People started to use fractal algorithms on the architect morphogenesis consciously after Mandel­brot’s publications. It was made possible to use fractal geometry for analyzing of architectural forms.

Fractals had become available to the majority of specialists due to the comput­erization.  They had been incredibly attractive for architectors, designers and town planners in aesthetic, philosophical and psychological way. Fractal theory was per­ceived on emotional, sensual level in the first phase. The constant repression lead­ing to loss of sensuality.

Application of fractal structures is effective on the microenvironment designing level: interior, household items and their elements. Fractal structures introduction allows creating a new surroundings for people with fractal properties on all levels. It corresponds to nesting spaces.

Fractal formations are not a panacea or a new era in the architect history. But it’s a new way to design architect forms which enriches the architectural theory and practice language. The understanding of na­ture fractal impacts on architectural view of urban environment. An attempt to de­velop the method of architectural designing which will base in an in-depth fractal forms is especially interesting. Will this method base only on mathematics? Will it be different methods and features symbiosis? The practice experiments and re­searches will show us. It’s safe to say that modern fractal approach can be useful not only for analysis, but also for harmonic order and nature’s chaos, architect which may be semantic dominant in nature and historic context.”

Computer systems. Fractal data compression is the most useful fractal applica­tion in the computer science. This kind of compression is based on the fact that it’s easy to describe the real world by fractal geometry. Nevertheless, pictures are compressed better than by other methods (like jpeg or gif). Another one advantage is that picture isn’t pixelateing while compressing. Often picture looks better after increase in fractal compressing.

Basic concept for fractal computer graphics is “Fractal triangle”. Also there are “Fractal figure”, “Fractal object”, “Fractal line”, “Fractal composition”, “Parent object” and “Heir object”. However, it should be noted that fractal computer graphics has recently received as a kind of computer graphics of 21th century.

 The opportunities of fractal computer graphics cannot be overemphasized. It allows creating abstract compositions where we can realize a lot of moves: hori­zontal and vertical, diagonal directions, symmetry and asymmetry etc. Only a few programmers from all over the world know about fractal graphics today.  To what can we compare fractal picture? For example, with complex structure of crystal or with snowflake, the elements of which line up in the one complex composition. This property of fractal object can be useful in ornament creating or designing of decorative composition. Algorithms of synthesis of fractal rates which allows to reproduce copy of any picture too close to the original are developed today.

III. Researching of computer programs of fractal construction

Strict algorithms of fractals are really good for programming. There are a lot of computer programs which introduce fractals nowadays. Computer mathematic systems are stand out from over programs, especially, Maple. Computer mathe­matics is mathematic modeling tool. So programming represents genetic structure of fractal in these systems and we can see precise submission of fractal structure in the picture while we enter a number of iterations . This is the reason why mathematic fractals should be studied with computer mathematics.  The last dis­covery in fractal geometry has been made possible by powerful, modern com­puters. Fractal property researching is almost completely based on computer cal­culations. It allows making computer experiments which reproduce processes and phenomenon which we can’t experiment in the real world with.

Our school has been worked with computer mathematics Maple package more than 10 years. So we have unique opportunity to experiment with mathematic fractals, thanks to that we can understand how initial values impact on outcome   (it is stochastic fractal). For example, we have understood the meaning of the fact that color is the fourth dimension: color changing leads to changing of physical char­acteristics. That is what astrophysics mean talking about “multicolored” of the Universe. While fractal constructing in interactive mathematic environment we re­ceived graphic models which was like A. D. Linde’s model of the Universe. Perhaps, it demonstrates that Universe has fractal structure.

Conclusion

Scientists and philosophers argue, can we talk about universality of fractals in recent years. There are two groups of two opposite positions. We agree with the fact that fractals are universal. Due to the fact that movement is inherent property of material also we always have fractals wherever we have movement.  

We are convinced that fractal is genetic property of the Universe, but it is not mean that all the Universe elements to the one fractal organization. In deployment process fractal structure is undergoing a lot of fluctuations (deviations) and a lot of points of bifurcation (branching) lead grate number of fractal development varie­ties.  

Therefore, we think that fractals are general academic method of real world re­searching. Fractals give the methodology of nature and community researching.

In transitional, chaotic period of society development social life become harder. Different social systems clash. Ancient values are exchanged for new values literally in all spaces. So it’s vitally important for science to develop behavior strategies which allow to avoid tragic mistakes. We think that fractals play important role in developing of such technologies. And – synergy is theory of evolving systems self- organization. But evolution happens on fractal principles, as we know now.

P.S.  Images - in attached files

Maple 2020 offers many improvements motivated and driven by our users.

Every single update in a new release has a story behind it. It might be a new function that a customer wants, a response to some feedback about usability, or an itch that a developer needs to scratch.

I’ll end this post with a story about acoustic guitars and how they drove improvements in signal and audio processing. But first, here are some of my personal favorites from Maple 2020.

Graph theory is a big focus of Maple 2020. The new features include more control over visualization, additional special graphs, new analysis functions, and even an interactive layout tool.

I’m particularly enamoured by these:

• We’ve introduced new centrality measures - these help you determine the most influential vertices, based on their connections to other vertices
• You now have more control over the styling of graphs – for example, you can vary the size or color of a nodebased on its centrality

I’ve used these two new features to identify the most influential MaplePrimes users. Get the worksheet here.

@Carl Love – looks like you’re the biggest mover and shaker on MaplePrimes (well, according to the eigenvector centrality of the MaplePrimes interaction graph).

We’ve also started using graph theory elsewhere in Maple. For example, you can generate static call graph to visualize dependencies between procedures calls in a procedure

You now get smoother edges for 3d surfaces with non-numeric values. Just look at the difference between Maple 2019 and 2020 for this plot.

Printing and PDF export has gotten a whole lot better.  We’ve put a lot of work into the proper handling of plots, tables, and interactive components, so the results look better than before.

For example, plots now maintain their aspect ratio when printed. So your carefully constructed psychrometric chart will not be squashed and stretched when exported to a PDF.

We’ve overhauled the start page to give it a cleaner, less cluttered look – this is much more digestible for new users (experienced users might find the new look attractive as well!). There’s a link to the Maple Portal, and an updated Maple Fundamentals guide that helps new users learn the product.

We’ve also linked to a guide that helps you choose between Document and Worksheet, and a link to a new movie.

New messages also guide new users away from some very common mistakes. For example, students often type “e” when referring to the exponential constant – a warning now appears if that is detected

We’re always tweaking existing functions to make them faster. For example, you can now compute the natural logarithm of large integers much more quickly and with less memory.

This calculation is about 50 times faster in Maple 2020 than in prior versions:

Many of our educators have asked for this – the linear algebra tutorials now return step by step solutions to the main document, so you have a record of what you did after the tutor is closed.

Continuing with this theme, the Student:-LinearAlgebra context menu features several new linear algebra visualizations to the Student:-LinearAlgebra Context Menu. This, for example, is an eigenvector plot.

Maple can now numerically evaluate various integral transforms.

The numerical inversion of integral transforms has application in many branches of science and engineering.

Maple is the world’s best tool for the symbolic solution of ODEs and PDEs, and in each release we push the boundary back further.

For example, Maple 2020 has improved tools for find hypergeometric solutions for linear PDEs.

This might seem like a minor improvement that’s barely worth mentions, but it’s one I now use all the time! You can now reorder worksheet tabs just by clicking and dragging.

The Hough transform lets you detect straight lines and line segments in images.

Hough transforms are widely used in automatic lane detection systems for autonomous driving. You can even detect the straight lines on a Sudoku grid!

The Physics package is always a pleasure to write about because it's something we do far better than the competition.

The new explore option in TensorArray combines two themes in Maple - Physics and interactive components. It's an intuitive solution to the real problem of viewing the contents of higher dimensional tensorial expressions.

There are many more updates to Physics in Maple 2020, including a completely rewritten FeynmanDiagrams command.

The Quantum Chemistry Toolbox has been updated with more analysis tools and curriculum material.

There’s more teaching content for general chemistry.

Among the many new analysis functions, you can now visualize transition orbitals.

I promised you a story about acoustic guitars and Maple 2020, didn’t I?

I often start a perfectly innocuous conversation about Maple that descends into several weeks of intense, feverish work.

The work is partly for me, but mostly for my colleagues. They don’t like me for that.

That conversation usually happens on a Friday afternoon, when we’re least prepared for it. On the plus side, this often means a user has planted a germ of an idea for a new feature or improvement, and we just have to will it into existence.

One Friday afternoon last year, I was speaking to a user about acoustic guitars. He wanted to synthetically generate guitar chords with reverb, and export the sound to a 32-bit Wave file. All of this, in Maple.

This started a chain of events that that involved least-square filters, frequency response curves, convolution, Karplus-Strong string synthesis and more. We’ll package up the results of this work, and hand it over to you – our users – over the next one or two releases.

Let me tell you what made it into Maple 2020.

Start by listening to this:

It’s a guitar chord played twice, the second time with reverb, both generated with Maple.

The reverb was simulated with convolving the artificially generated guitar chord with an impulse response. I had a choice of convolution functions in the SignalProcessing and AudioTools packages.

Both gave the same results, but we found that SignalProcessing:-Convolution was much faster than its AudioTools counterpart.

There’s no reason for the speed difference, so R&D modified AudioTools:-Convolution to leverage SignalProcessing:-Convolution for the instances for which their options are compatible. In this application, AudioTools:-Convolution is 25 times faster in Maple 2020 than Maple 2019!

We also discovered that the underlying library we use for the SignalProcessing package (the Intel IPP) gives two options for convolution that we were previously not using; a method which use an explicit formula and a “fast” method that uses FFTs. We modified SignalProcessing:-Convolution to accept both options (previously, we used just one of the methods),

That’s the story behind two new features in Maple 2020. Look at the entirety of what’s new in this release – there’s a tale for each new feature. I’d love to tell you more, but I’d run out of ink before I finish.

To read about everything that’s new in Maple 2020, go to the new features page.

Today we celebrated International Women's Day at Maplesoft. As part of our celebration, we had a panel of 5 successful women from within the community share their experiences and insights with us. 

Hearing these women speak has given me the courage to share my personal experience and advice to women in technology. If what I write here helps even one woman, then I will have accomplished something great today. 

-----

What do you do at Maplesoft?

My name is Karishma. I'm the Director, Product Management - Academic. 

Where did you grow up and where did you go to school (Diploma/degree)?

I was born and raised in Montreal to parents of Indian descent. Like most Indian parents, they “encouraged" me to pursue a career in either Law, Medicine, or Engineering, despite my true calling to pursue a career in theatre (at least that's what I believed it to be at the time). 

Given that I had no siblings to break the ice, and that rebelling wasn't my Modus Operandi (that came much later), I did what any obedient teenager would do: I pursued a career in Electrical Engineering at McGill University. In my mind, this was the fastest way to landing a job and fleeing the proverbial nest. 

Electrical Engineering was far from glamorous, and after two years, I was ready to switch. It was due to the sheer insistence of my mother that I completed the degree. 

So how did I end up pursuing a graduate degree in Biomedical Engineering at McGill University? It wasn't the future I envisioned, but the economic downturn in 2001-2002 saw a massive decrease in hiring, and the job that I had held-out patiently for during those four years became a far-off dream. So I did the thing I never imagined I would: I accepted the offer to pursue a Master's and the very generous stipend that came with it. In case you are wondering, I only applied because my father nagged me into submission. (Insistence and nagging are two innate traits of Indian parents). 

Contrary to what I expected, I loved my Master's degree! It gave me the freedom to immerse myself wholly in a topic I found exciting and allowed me to call the shots on my schedule, which led to my involvement in student government as VP Internal. But apart from the research and the independence, pursuing a master's degree opened doors to opportunities that I couldn't have imagined, such as an internship with the International Organization for Migration in Kenya, a job offer in Europe, and the chance to work at Maplesoft. (I guess my parents did know what was best for me.) 

What is the best part of your job?

It's figuring out how to solve problems our users have as well as the ones they might not realize they have. 

At Maplesoft, I work with some most brilliant minds I've ever encountered to build a product that makes math more accessible to our users, whether they be a student, researcher, scientist, or engineer. 

Some of the aspects of my role that I love the most include: 

• speaking to and learning from our customers, 
• interpreting the meaning behind their words, facial expressions, vocal intonations, and body language, and
• collaborating with the sales, marketing, and development teams to turn what was 'said' into tangible actions that will enhance the product and user experience. 

Most nights, when I leave work, I do so with a sense of excitement because I know my actions and the actions of those I work with will help our users achieve their goals and ambitions. There's no better high. 

What advice do you have for young women interested in a career in your field? 
Throughout my career, I've had the privilege to work with some amazing women and men who've given me advice that I wish I had known when I was an undergraduate student. If you are a woman pursuing a STEM degree or starting your first job in a tech firm, here are three tips that may help you: 

1.   Don't be scared of the 'N' word. 
Don't be scared of NETWORKING. I know it can be intimidating, but it truly is the best way to land a job, advance your career, or meet the person you admire most. Remember that networking can take place anywhere - it's not exclusive to networking events. Some advice that I received that helped me overcome my fear of networking: 

• Smile - Before you approach a person or enter a networking session, force yourself to smile. It will help you diffuse any tension you are holding and will make you appear more approachable. 
• Research - Take the time to research the person(s) you would like to meet. Find out as much as you can about them and their company. Prepare some icebreaker questions and other questions to help carry the conversation forward ahead of time. Remember that people like to talk about themselves and their experiences. 
• Don't take it personally - The person you approach may find networking equally tricky. So if they seem disinterested or aloof, don't take it personally. 
• Just do it - Networking gets easier with practice. Don't let a failed attempt set you back. The worse thing that will happen is that you don't make a connection. 

 2.   It's ok to ask for help.
If you are a woman in an environment that is dominated by men, you might hesitate to ask for help. DON'T! There's nothing wrong with asking for help. That said, many women ask for help in a way that undermines their confidence and thus erodes others’ perception of them. Next time you need help, have a question or require clarification, take a moment to phrase your request, so you don't inadvertently put yourself down. 

3.   Play to your strengths
Don't think you need to know everything. Nobody expects it. If you landed a new job or co-op placement, and you are finding yourself doing things you've never done and don't come naturally to you yet, don't let your brain convince you that you don't deserve it. Remember that you earned it because of your qualities and strengths. 

Playing mini-golf recently, I realized that my protractor can only help me so far since it can't calculate the speed of the swing needed.  I decided a more sophisticated tool was needed and modeled a trick-shot in MapleSim.

To start, I laid out the obstacles, the ball and club, the ground, and some additional visualizations in the MapleSim environment.

When running the simulation, my first result wasn't even close to the hole (similar to when I play in real life!).

The model clearly needed to be optimized. I went to the Optimization app in MapleSim (this can be found under Add Apps or Templates  on the left hand side).

Inside the app I clicked "Load System" then selected the parameters I wanted to optimize.

For this case, I'm optimizing 's' (the speed of the club) and 'theta' (the angle of the club). For the Objective Function I added a Relative Translation Sensor to the model and attached a probe to the Vector Norm of the output.

Inside the app, I switched to the Objective Function section.  Selecting Probes, I added the new probe as the Objective Function by giving it a weight of 1.

Scrolling down to "Execute Parameter Optimization", I checked the "Use Global Optimization Toolbox" checkbox, and clicked Run Parameter Optimization.

Following a run time of 120 seconds, the app returns the graph of the objective function. 

Below the plot, optimal values for the parameters are given. Plugging these back into the parameter block for the simulation we see that the ball does in fact go into the hole. Success!

Mini_golf_Global_Optimization.msim

We recently had a question about using some of the plotting commands in Maple to draw things. We were feeling creative and thought why not take it a step further and draw something in 3D.

Using the geom3d, plottools, and plots packages we decided to make a gingerbread house.

To make the base of the house we decided to use 2 cubes, as these would give us additional lines and segments for the icing on the house.

point(p__1,[2,3,2]):
point(p__2,[3,3,2]):
cube(c1,p__1,2):
cube(c2,p__2,2):
base:=draw([c1,c2],color=tan);

Using the same cubes but changing the style to be wireframe and point we made some icing lines and decorations for the gingerbread house.

base_decor1:=draw([c1,c2],style=wireframe,thickness=3,color=red,transparency=0.2):
base_decor2:=draw([c1,c2],style=wireframe,thickness=10,color=green,linestyle=dot):
base_decor3:=draw([c1,c2],style=point,thickness=2,color="Silver",symbol=sphere):
base_decor:=display(base_decor1,base_decor2,base_decor3);

To create the roof we found the vertices of the cubes and used those to find the top corners of the base.

v1:=vertices(c1):
v2:=vertices(c2):
pc1:=seq(point(pc1||i,v1[i]),i=1..nops(v1)):
pc2:=seq(point(pc2||i,v2[i]),i=1..nops(v2)):
topCorners:=[pc1[5],pc1[6],pc2[1],pc2[2]]:
d1:=draw(topCorners):

Using these top corners we found the midpoints (where the peak of the roof would be) and added the roof height to the coordinates.

midpoint(lc1,topCorners[1],topCorners[2]):
detail(lc1);

point(cc1,[-(2*sqrt(3))/3 + 2, (2*sqrt(3))/3 + 3+1, 2]):
d3:=draw(cc1):

midpoint(lc2,topCorners[3],topCorners[4]):
detail(lc2);

point(cc2,[(2*sqrt(3))/3 + 3, (2*sqrt(3))/3 + 3+1, 2]):
d4:=draw(cc2):

With the midpoints and vertices at the front and rear of the house we made two triangles for the attic of the gingerbread house.

triangle(tf,[topCorners[1],topCorners[2],cc1]):
front:=draw(tf,color=brown):

triangle(tb,[topCorners[3],topCorners[4],cc2]):
back:=draw(tb,color=tan):

Using these same points again we made more triangles to be the roof.

triangle(trl,[cc1,cc2,pc1[5]]):
triangle(trh,[pc2[2],pc1[6],cc1]):
triangle(tll,[cc1,cc2,pc2[2]]):
triangle(tlh,[pc2[1],pc1[5],cc2]):
roof:=draw([trl,trh,tll,tlh],color="Chocolate");

Our gingerbread house now had four walls, a roof, and icing, but no door. Creating the door was as easy as making a parallelepiped, but what is a door without more icing?

door:=display(plottools:-parallelepiped([1,0,0],[0,1.2,0],[0,0,0.8],[0.8,1.9,1.6]),color="DarkRed"):
door_decor1:=display(plottools:-parallelepiped([1,0,0],[0,1.2,0],[0,0,0.8],[0.8,1.9,1.6]),color="Gold",style=line):
door_decor2:=display(plottools:-parallelepiped([1,0,0],[0,1.2,0],[0,0,0.8],[0.8,1.9,1.6]),color="Silver", style=line,linestyle=dot,thickness=5):
door_decor:=display(door_decor1,door_decor2):

Now having a door we could have left it like this, but what better way to decorate a gingerbread house than with candy canes? Naturally, if we were going to have one candy cane we were going to have lots of candy canes. To facilitate this we made a candy cane procedure.

candy_pole:=proc(c:=[0,0,0], {segR:=0.1}, {segH:=0.1}, {segn:=7}, {tilt_theta:=0}, {theta:=0}, {arch:=true}, {flip:=false})
local cane1,cane2,cane_s,cane_c,cane0,cane,i,cl,cd,ch, cane_a,tmp,cane_ac,cane_a1,cane00,cane01,cane02,cane_a1s,tmp2,cane_a2s:
uses plots,geom3d:
cl:=c[1]:
cd:=c[2]:
ch:=c[3]:
cane1:=plottools:-cylinder([cd, ch, cl], segR, segH,style=surface):

cane2:=display(plottools:-rotate(cane1,Pi/2,[[cd,ch,cl],[cd+1,ch,cl]])):
cane_s:=[cane2,seq(display(plottools:-translate(cane2,0,segH*i,0)),i=1..segn-1)]:
cane_c:=seq(ifelse(type(i,odd),red,white),i=1..segn):

cane0:=display(cane_s,color=[cane_c]):

if arch then

cane_a:=plottools:-translate(cane2,0,segH*segn-segH/2,0):
tmp:=i->plottools:-rotate(cane_a,i*Pi/24, [ [cd,ch+segH*segn-segH/2,cl+segR*2] , [cd+1,ch+segH*segn-segH/2,cl+segR*2] ] ):

cane_ac:=seq(ifelse(type(i,odd),red,white),i=1..24):

                cane_a1s:=seq(plottools:-translate(tmp(i),0,segH*i/12,segR*i/4),i=1..12):

tmp2:=i->plottools:-rotate(cane_a1s[12],i*Pi/24,[[cd,ch+segH*segn-0.05,cl+segR*2],[cd+1,ch+segH*segn-0.05,cl+segR*2]]):

cane_a2s:=seq(plottools:-translate(tmp2(i),0,-segH*i/500,segR*i/4),i=1..12):
cane_a1:=display(cane_a1s,cane_a2s,color=[cane_ac]):
cane00:=display(cane0,cane_a1);

                if flip then

cane01:=plottools:-rotate(cane00,tilt_theta,[[cd,ch,cl],[cd+1,ch,cl]]):
cane02:=plottools:-rotate(cane01,theta,[[cd,ch,cl],[cd,ch+1,cl]]):
cane:=plottools:-reflect(cane01,[[cd,ch,cl],[cd,ch+1,cl]]):

                else

cane01:=plottools:-rotate(cane00,tilt_theta,[[cd,ch,cl],[cd+1,ch,cl]]):
cane:=plottools:-rotate(cane01,theta,[[cd,ch,cl],[cd,ch+1,cl]]):

                end if:

                return cane:

else

                cane:=plottools:-rotate(cane0,tilt_theta,[[cd,ch,cl],[cd+1,ch,cl]]):

                return cane:

end if:

end proc:

With this procedure we decided to add candy canes to the front, back, and sides of the gingerbread house. In addition we added two candy poles.

Candy Canes in front of the house:

cane1:=candy_pole([1.2,0,2],segn=9,arch=false):
cane2:=candy_pole([2.8,0,2],segn=9,arch=false):
cane3:=candy_pole([2.7,0.8,3.3],segn=9,segR=0.05,tilt_theta=-Pi/7):
cane4:=candy_pole([1.3,0.8,3.3],segn=9,segR=0.05,tilt_theta=-Pi/7,flip=true):
front_canes:=display(cane1,cane2,cane3,cane4):

Candy Canes at the back of the house:

caneb3:=candy_pole([1.5,4.2,2.5],segn=15,segR=0.05,tilt_theta=-Pi/3,flip=true):
caneb4:=candy_pole([2.5,4.2,2.5],segn=15,segR=0.05,tilt_theta=-Pi/3):}
back_canes:=display(caneb3,caneb4):

Candy Canes at the left of the house:

canel1:=candy_pole([0.8,1.5,2.5],segn=15,segR=0.05,tilt_theta=-Pi/7,theta=Pi/2):
canel2:=candy_pole([0.8,2.5,2.5],segn=15,segR=0.05,tilt_theta=-Pi/7,theta=Pi/2):
canel3:=candy_pole([0.8,4,2.5],segn=15,segR=0.05,tilt_theta=-Pi/7,theta=Pi/2):
left_canes:=display(canel1,canel2,canel3):

Candy Canes at the right of the house:

caner1:=candy_pole([3.2,1.5,2.5],segn=15,segR=0.05,tilt_theta=-Pi/7,theta=Pi/2):
caner2:=candy_pole([3.2,2.5,2.5],segn=15,segR=0.05,tilt_theta=-Pi/7,theta=Pi/2):
caner3:=candy_pole([3.2,4,2.5],segn=15,segR=0.05,tilt_theta=-Pi/7,theta=Pi/2):
right_canes:=display(caner1,caner2,caner3):

canes:=display(front_canes,back_canes,right_canes,left_canes):

With these canes in place all that was left was to create the ground and display our Gingerbread House.

ground:=display(plottools:-parallelepiped([5,0,0],[0,0.5,0],[0,0,4],[0,1.35,0]),color="DarkGreen"):

display([door,door_decor,d1,base,base_decor,d3,d4,front,back,roof,ground,canes],orientation=[-100,0,95]);

You can download the full worksheet creating the gingerbread house below:

Geometry_Gingerbread.mw

I'm only just hearing (haven't experienced) about some serious issues with the 2019.2 updates.  I would recommend waiting for Maplesoft to release an emergency 2019.3 fix update - Maplesoft can NOT leave the last update of 2019 in this state.

Hi there! 

One of my favorite videogames is pokémon as you can probably guess from the title. As a player I always wanted to optimize my chances of obtaining the rarest and best pokémon in the game. I have been working on an application that aims to use graph theory to analyze the game Pokémon Blue. The application explores the following questions:

Which is the rarest pokémon in the game?
Where can I find an specific pokémon and with what probabilities?
What is the place with most different species of wild pokémon?

I also included algorithms for the following: Given a certain desired team

• Find the minimum amount of places to visit to catch them and return the list of the places the player will need to visit.
• What are the routes with best probabilities to catch each pokémon from my desired team?

Check out my application at: https://www.maplesoft.com/applications/view.aspx?SID=154565.

The following are some of the results obtained in the app:

What is the most common pokémon?

I did not only considered the amount of places a pokémon can appear in but also the probabilities of it appearing in each place.

What are the connections between pokémon and places?

In my graph, I connected a pokémon and a place if such pokémon could be caught in that place. The following is an example for the pokémon Pidgey. The weights of the edges are the probabilities of finding Pidgey in each route.

Viceversa, I did the same for how a route is connected to the pokémons in it: