## Gingerbread House with 3D Geometry

Maple 2019

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:

## Wait for 2019.3 - 2019.2 flawed - Maplesoft please...

Maple 2019

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.

## Graph Theory and PokÃ©mon Application

by: Maple 2019

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: