Archive for the ‘geometry’ Category

h1

2. Middle School Madness: BIG MATHS

June 1, 2017

Mathspig just loves this Fluid Geometry Mural by artist Clint Fulkerson. Here is a Speed Clip of Flulkerson at the University of Maine Art Gallery 2012.

Mathspig  went crazy trying to produce her own Fluid Geometry Mural. These guys are artists. But your middle school maths class can use a simple method (See GIF below) and make a fab mural on a white/black board or down a school hallway. This is a great end of year/semester/pre-holidays project.

The reason for tackling such a project is to engage students, who are otherwise not interested in maths as well as do some whole school PR for maths. The mural WILL be noticed. And it makes maths look COOL.You will find more interesting Fluid Geometry works by Clint Fulkerson here.

Advertisements
h1

CGI and Middle School Maths … Metric units

April 9, 2017

1. Story Board Sketches

The Pixar film, Finding Dory, began as storyboard sketches. There were 103,000 drawn for the film. The sketches are placed side-by-side in sequence in order to convey scenes and deliver a rough sense of how the story unfolds.

How many 64-page Exercise books would the story board take?

Only one side of the page of the 64-page exercise book would be used for storyboarding.

No. exercise books = 102,000/32  =  3,187.5

If put into flip Book form the story board sketches would produce a ‘rough’ version of the film.

Superman Vs Goku

………………………………………..

2. Ratio

If you are making a film about, say, Godzilla, you want him to be big. VERY BIG. But he has to share a screen with human characters.

If Godzilla is too big all the humans look like ants.

If he is too small … he looks like a toy TRex.

The Monster Maths:

Height: 108.2m Godzilla’s towering height in the 2014 film—the tallest onscreen incarnation ever

Tail: 167.74m Total length of Godzilla’s spiked tail

Volume: 89,724 m3 Godzilla’s total volume in the 2014 film.

Volume: 90,000 tons Godzilla’s volume if filled with water

Teeth: 1.07m  Length from the root to the tips of Godzilla’s canine teeth

Teeth: 60 Teeth in Godzilla’s mouth

Roar: 4.83km Approximate distance Godzilla’s roar reverberates. (100,000W Power of the 3.7m-high, 5.5m-wide speaker array from which the sound designers blasted Godzilla’s roar to record the sound in a “real world” context)

Feet: 17.66m  Total width of Godzilla’s feet across the widest point

Feet: 18.18m Length of Godzilla’s footprint from toe to heel

You vs Godzilla

The average height for 14-year-old boys in the U.S. is 164 cm.

For 14-year-old girls in the U.S. the average height is 158.75 cm.

Godzilla = 108.2 m  

We’ll assume an average height of, say, 1.6m for a 14 yo.

14 yo =  1.6 m    

Height:

108.2/1.6 = 67.6 = 68     

Footprint

You vs Godzilla’s Footprint

18.18/160 =  11.4         

Teeth

You vs Godzilla’s Tooth

1.07/1.60 = 0.7                      

………………………………………………

3. 3D Modeling: Polygon Count

3D modeling can be approached in several ways. There is a good explanation here.

One method is to use a 3D scanner to produce a 3D mesh.

Godzilla model under construction

Digital Godzilla as 3D mesh

The mesh consists of polygons. You’ll find everything you want to know about polygons here.

The greater the no. of polygons the better the resolution in a 3D mesh, the better the  resolution of the character or object as more shading and texture variation can be added.

So the POLYGON COUNT or POLY COUNT is a measure of the quality of an image.

Here are some POLY COUNTS:

Godzilla

500,000 polygons.

It took the 4 CGI artists 6 months to fully nail the texture of Godzilla’s scales. (See above)

Elysium

Just the space station ran close to 3 trillion polygons.

Transformers 2

Devastator had 13 million polygons

Clash of the Titans

The Kraken had 7 million Polygons.

Avatar

Apart from some bluescreen shots of live-action actors in cockpits, it’s all CG: gunships, missiles, smoke trails, water, fire, an army of photorealistic virtual characters and a giant tree made of 20 million polygons with 1.2 million leaves.

More here.

Lord of the Rings – The Two Towers (2002)

(See below)

The Gollum head model consisted of just over 2,600 polygons, which were mostly quads.

More here.

Film Animation vs Game Animation

The POLY COUNT is always greater in animated movies than in video games mainly because a higher poly count takes longer and costs more. A higher resolution is needed in a movie as it is shown on a much bigger screen.

Gollum: Game vs Film 3D model

Here are some POLY COUNTS for some popular games:

The Amazing Spiderman

Spiderman – 11,652

Assassin’s Creed 3

Benjamin Franklin – 17,744

Charles Lee – 25,994

Connor Kenway – 28,501

Desmond – 14,934

Haytham Kenya – 19,985

Batman: Arkham City

Batman: Film Version

Batman: Game Model

Batman Beyond suit – 13,050

Harley Quinn – 17,731

Harley Quinn (DLC) – 19,110

Poison Ivy – 15,977

Call of Duty: Black Ops

Frank Woods (full gear) – 19,777

Crysis 2 (console)

Nano suit – 19,073

Multiplayer suit – 27,414

………………….

FREE 3D MODELING FOR MIDDLESCHOOL STUDENTS

TINKERCAD is the free online resource that allows students to create their own 3D graphs – like Godzilla vs You (above) or 3D images that can be used for 3D printing. Play with the program. Mathspig had a little play and produced the genetically challenged chicken (below).

Tinkercad Dashboard

Find here.

Drag and drop shapes.

More here.

Play with shape dimensions.

More here.

Create figures for 3D printer.

MATHSPIG REPORT CARD: More playing with program needed!!!

h1

CGI and Middle School Math …. USA units

April 9, 2017

1. Story Board Sketches

The Pixar film, Finding Dory, began as storyboard sketches. There were 103,000 drawn for the film. The sketches are placed side-by-side in sequence in order to convey scenes and deliver a rough sense of how the story unfolds.

How many 64-page Exercise books would the story board take?

Only one side of the page of the 64-page exercise book would be used for storyboarding.

No. exercise books = 102,000/32  =  3,187.5

If put into flip Book form the story board sketches would produce a ‘rough’ version of the film.

Superman Vs Goku

………………………………………..

2. Ratio

If you are making a film about, say, Godzilla, you want him to be big. VERY BIG. But he has to share a screen with human characters.

If Godzilla is too big all the humans look like ants.

If he is too small … he looks like a toy TRex.

The Monster Maths:

Height: 355ft  Godzilla’s towering height in the 2014 film—the tallest onscreen incarnation ever

Tail: 550ft 4in Total length of Godzilla’s spiked tail.

Volume: 90,000 tons Godzilla’s volume if filled with water

Teeth: 3.51ft Length from the root to the tips of Godzilla’s canine teeth

Teeth: 60 Teeth in Godzilla’s mouth

Roar: 3 miles. Approximate distance Godzilla’s roar reverberates. (100,000W Power of the 12-foot-high, 18-foot-wide speaker array from which the sound designers blasted Godzilla’s roar to record the sound in a “real world” context)

Feet: 58ft  Total width of Godzilla’s feet across the widest point

Feet: 60ft  Length of Godzilla’s footprint from toe to heel

You vs Godzilla

The average height for 14-year-old boys in the U.S. is 64.5 inches or 5 feet 4½ inches.

For 14-year-old girls in the U.S. the average height is 62.5 inches or 5 feet 2½ inches.

Godzilla = 355ft

14 yo =  5’ 3”

Height:

355/5.25= 67.6 = 68

To make a 3D graph like this see below.

Footprint

You vs Godzilla’s Footprint

60/5.25 =  11.4

Teeth

You vs Godzilla’s Tooth

 3.5/5.25 = 0.7

………………………………………………

3. 3D Modeling: Polygon Count

3D modeling can be approached in several ways. There is a good explanation here.

One method is to use a 3D scanner to produce a 3D mesh.

Godzilla model under construction

Digital Godzilla as 3D mesh

The mesh consists of polygons. You’ll find everything you want to know about polygons here.

The greater the no. of polygons the better the resolution in a 3D mesh, the better the  resolution of the character or object as more shading and texture variation can be added.

So the POLYGON COUNT or POLY COUNT is a measure of the quality of an image.

Here are some POLY COUNTS:

Godzilla

500,000 polygons.

It took the 4 CGI artists 6 months to fully nail the texture of Godzilla’s scales. (See above)

Elysium

Just the space station ran close to 3 trillion polygons.

Transformers 2

Devastator had 13 million polygons

Clash of the Titans

The Kraken had 7 million Polygons.

Avatar

Apart from some bluescreen shots of live-action actors in cockpits, it’s all CG: gunships, missiles, smoke trails, water, fire, an army of photorealistic virtual characters and a giant tree made of 20 million polygons with 1.2 million leaves.

More here.

Lord of the Rings – The Two Towers (2002)

(See below)

The Gollum head model consisted of just over 2,600 polygons, which were mostly quads.

More here.

Film Animation vs Game Animation

The POLY COUNT is always greater in animated movies than in video games mainly because a higher poly count takes longer and costs more. A higher resolution is needed in a movie as it is shown on a much bigger screen.

Gollum: Game vs Film 3D model

Here are some POLY COUNTS for some popular games:

The Amazing Spiderman

Spiderman – 11,652

Assassin’s Creed 3

Benjamin Franklin – 17,744

Charles Lee – 25,994

Connor Kenway – 28,501

Desmond – 14,934

Haytham Kenya – 19,985

Batman: Arkham City

Batman: Film Version

Batman: Game Model

Batman Beyond suit – 13,050

Harley Quinn – 17,731

Harley Quinn (DLC) – 19,110

Poison Ivy – 15,977

Call of Duty: Black Ops

Frank Woods (full gear) – 19,777

Crysis 2 (console)

Nano suit – 19,073

Multiplayer suit – 27,414

………………….

FREE 3D MODELING FOR MIDDLESCHOOL STUDENTS

TINKERCAD is the free online resource that allows students to create their own 3D graphs – like Godzilla vs You (above) or 3D images that can be used for 3D printing. Play with the program. Mathspig had a little play and produced the genetically challenged chicken (below).

Tinkercad Dashboard

Find here.

Drag and drop shapes.

More here.

Play with shape dimensions.

More here.

Create figures for 3D printer.

MATHSPIG REPORT CARD: More playing with program needed!!!

h1

Smoke Jumpers and their Amazing Wildfire Maths . . . . . . . . . . . . . . METRIC UNITS

December 5, 2016

mathspig-smoke-jumpers 

With wildfires in Texas and now Australia is facing the fire season, it is time to think about fire fighter maths.

mathspig-metric-units-fire-math

METRIC UNITS

Background Story

On 5th August 1949 Wag Dodge was dropped by parachute with 14 other fire fighters into Mann Gulch, a steep-sided gully in a Montana pine forest. Fire fighters who parachute in to put out small blazes started by lightening are called Smoke Jumpers. As they worked their way down the sides of the gully the breeze was blowing away from them. But the wind soon shifted. This produced an updraft, which increases the speed of the fire front. The 15 Smoke Jumpers turned and started running for their lives uphill.

HOW FAST CAN YOU RUN?

METRIC UNITS

Time Trial:

Mark out a 10 m course. Make 3 time trials.

t1 =

t2 =

t3=

Average your time:

tav = (t1 + t2 + t3)/ 3 =

Your Speed S = 10/tav = ……… m/sec

mathspig-firefirghter-maths-1

HOW FAST IS A GRASS FIRE?

This will, of course, vary depending on the wind speed. A typical grass fire in Australia in a flat area can travel at 20kph (up to 30 kph) in a gentle breeze.

Fire Front Speed Grass Fire

Fire Front Speed = 20 kph = 20 x1000/(60 x 60)

                               = 20 x 0.27777777 = 20 x 0.28 m/sec

                               = 5.6 m/sec

mathspig-firefighter-maths-2

CAN YOU OUT RUN A FIRE?

Average Running Speed Boy 13–14 yo = 3.0 m/sec

Average Running Speed Girl 13–14 yo = 2.4 m/sec

We’ll assume, boy or girl, that you are really motivated and can run away from the fire at top speed of 3.0 m/sec. Now calculate the distance you can run and the fire front moves in 10 secs intervals up to 1 minute.

mathspig-fire-fighter-table-1

This is not looking good. See more Firefighters Need Maths here.

We can do very accurate calculations using simultaneous equations. Wildfire Algebra: Detailed Worksheet using simultaneous equations and solutions  here.

NOW YOU ARE RUNNING UP HILL. WHAT HAPPENS?

We’ll assume, due to being motivated by having a fire licking your heels, that you can run at your top speed up hill for a short time, at least. But here is the problem.

Heat rises and so there is a Chimney Effect pushing the fire uphill. The rule of thumb used by fire fighters is:

Each 10º increase in slope, the fire front speed doubles.

mathspig-cfa-diag

mathspig-fire-fighter-table-2

Now you can calculate the distance travelled by the fire front up a slope at a 30º angle.

Don’t forget you can use the WEB 2.0 Calculator here.

mathspig-fire-fighter-table-3

Even at your top running speed, which is unlikely up a slope, you can run 180 m in 1 minute. In that time the forefront has moved 2688 m or 2.7 km.

It depends how far away you are from the fire front, but it seems you cannot out run this fire front.

Again we can do very accurate calculations using simultaneous equations.

See Firefighters Need Maths here.

Wildfire Algebra: Worksheet and solutions here.

CAN YOU OUT RUN A WILD FIRE?

High winds can turn a bush or forrest fire into a WILD FIRE with wind speeds up to 110 kph and temperatures up to 2000 °C, which can and does melt glass and cars.

The fire front speed doubles with every 10º, so speeds for the fire front can reach 220 kph, 330kph and up to 550kph.

20o-angle-mathspig-2

What happened to the Smoke Jumpers?

When the fire front changed direction Wag Dodge and 14 other Smoke Jumpers found themselves running for their lives up a steep slope. What did Wag do next?

ANS: Here’s the amazing thing. Wag realised he could not out run the fire at that point. So he stopped. Took off his back pack. Took out some MATCHES and lit a fire in the grassy patch in front of him. Just before the firewall hit he threw himself face down on the burnt patch. He survived. The other 14 firefighters did not.

h1

Smoke Jumpers and their Amazing Wildfire Maths . . . . . . . . . . . . . . USA UNITS

December 5, 2016

 

mathspig-smoke-jumpers 

With wildfires in Texas and now Australia is facing the fire season, it is time to think about fire fighter maths.

mathspig-usa-units-fire-math

 

USA UNITS

Background Story

On 5th August 1949 Wag Dodge was dropped by parachute with 14 other fire fighters into Mann Gulch, a steep-sided gully in a Montana pine forest. Fire fighters who parachute in to put out small blazes started by lightening are called Smoke Jumpers. As they worked their way down the sides of the gully the breeze was blowing away from them. But the wind soon shifted. This produced an updraft, which increases the speed of the fire front. The 15 Smoke Jumpers turned and started running for their lives uphill.

HOW FAST CAN YOU RUN?

USA UNITS

Time Trial:

Mark out a 30ft course. Make 3 time trials.

t1 =

t2 =

t3=

Average your time:

tav = (t1 + t2 + t3)/ 3 =

Your Speed S = 30/tav   ft/sec

mathspig-firefirghter-maths-1

HOW FAST IS A GRASS FIRE?

This will, of course,  vary depending on the wind speed. A typical grass fire in Australia in a flat area can travel at 12mph (up to 20mph) in a gentle breeze.

Fire Front Speed Grass Fire

Fire Front Speed = 12 mph = 12 x 5280/(60 x 60)

                           = 17.6 ft/sec

                           = 18 ft/sec

mathspig-firefighter-maths-2

CAN YOU OUT RUN A FIRE?

Average Running Speed Boy 13–14 yo = 10 ft/sec

Average Running Speed Girl 13–14 yo = 8 ft/sec

We’ll assume, boy or girl, that you are really motivated and can run away from the fire at top speed of 10 ft/sec and -Wow! – this is easy math. Now calculate the distance you can run and the fire front moves in 10 secs intervals up to 1 minute.

mathspig-fire-fighter-table-4

This is not looking good. See more Firefighters Need Maths here.

We can do very accurate calculations using simultaneous equations. Wildfire Algebra: Detailed Worksheet using simultaneous equations and solutions  here.

NOW YOU ARE RUNNING UP HILL. WHAT HAPPENS?

mathspig-cfa-diag-usa-units

We’ll assume, due to being motivated by having a fire licking your heels, that you can run at your top speed up hill for a short time, at least. But here is the problem.

Heat rises and so there is a Chimney Effect pushing the fire uphill. The rule of thumb used by fire fighters is:

Each 10º increase in slope, the fire front speed doubles.

mathspig-fire-fighter-table-5

Now you can calculate the distance travelled up a slope at a 30º angle.

Don’t forget you can use the WEB 2.0 Calculator here

mathspig-fire-fighter-table-6

Even at your top running speed, which is unlikely up a slope, you can run 1080 ft in 1 minute. In that time the forefront has moved 8640 ft or 1.6 miles. It depends how far away you are from the fire front when you start running, but it seems likely that you cannot out run this fire front.

Again we can do very accurate calculations using simultaneous equations.

See Firefighters Need Maths here.

Wildfire Algebra Worksheet and solutions  here.

CAN YOU OUT RUN A WILD FIRE?

High winds can turn a bush or forrest fire into a WILD FIRE with wind speeds up to 70 mph and temperatures up to 2000 °C, which can and does melt glass and cars.

The fire front speed doubles with every 10º, so speeds for the fire front in a strong wind can reach 140 mph, 210 mph and up to 280 mph.

20o-angle-mathspig-2

What happened to the Smoke Jumpers?

When the fire front changed direction Wag Dodge and 14 other Smoke Jumpers found themselves running for their lives up a steep slope. What did Wag do next?

ANS: Here’s the amazing thing. Wag realised he could not out run the fire at that point. So he stopped. Took off his back pack. Took out some MATCHES and lit a fire in the grassy patch in front of him. Just before the firewall hit he threw himself face down on the burnt patch. He survived. The other 14 firefighters did not.

h1

FOOTBALL MATHS: THE LONGEST KICK

June 3, 2016

Mathspig Football Maths 1

Mathspig football Maths 1a

Mathspig Football Maths 2aMathspig Football Maths 3Mathspig Football Maths 4

………………………………………………

The following maths is suitable for Year 9+

but can be presented to lower grades just to show

maths is cool!

Mathspig Football Maths 5Mathspig Football Maths 6bMathspig Football Maths 7

h1

MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig MathsPig …………………………………………………. with Kerry Cue

March 1, 2016

 

1. MATHSPIG CUBISTHellooooooooo My Sweet Little Picassos,

Mathspig has gone, like, totally Cubist this month. You too can turn any portrait of yourself into a Cubist Master piece (See Mathspig portrait above), here.

Mathspigs maths friends, Lyn and Erwin, who I met at the 13e Salon Culture & Jeux Mathematique, Paris, have sent me a reminder about their amazing Cryptocube construction kit (Mathspig is twirling one above) . This is not for the faint hearted. It’s a Big Maths challenge, but well worth the investment especially for schools. You can learn more about the Zometool Cryptocube construction kit here.

Meanwhile, here are links to 10 Amazing Ways to See a Cube:

1. Tube Cube

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

2. Folded Paper Cube

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

3. Anamorphic Cube

>>>>>>>>>>>>>>>>>>>>>>>>

4. Floating Cube

>>>>>>>>>>>>>>>>>>>>>>>>

5. Street Art Cube

>>>>>>>>>>>>>>>>>>>>>>>>

6. EDIBLE CUBE

>>>>>>>>>>>>>>>>>>>>>>>>

7. GIANT CUBE

>>>>>>>>>>>>>>>>>>>>>>>>

8. Spaghetti Cube

>>>>>>>>>>>>>>>>>>>>>>>>

9. Fashion Cube

>>>>>>>>>>>>>>>>>>>>>>>>

10. CUBIC BUBBLE

…………………………………………………………….

Toodles

…………………………………………………….

Mathspig