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Physics, Astronomy & Cosmology / Re: Could the universe be smaller than the visible universe?

« on: Yesterday at 01:06:13 »

Einstein's equations gave two solutions

A brief history of modern cosmology: Around 1915, Einstein produced a field equation describing the cosmos. The calculation generated an (unknown) constant of integration (as integration always does).
- Einstein's original paper omitted mention of this constant. It was mentioned in a later version.
- As usual in mathematics, you try to find some known condition that allows you to identify the value of this integration constant
- Einstein followed the cosmological assumption of his day that the universe was eternal and unchanging, and this "known condition" allowed him to define a value for this constant.
- Around 1925, Belgian priest Georges Lemaitre (perhaps with a Bible-inspired cosmology) saw in these equations the possibility of what we now call a "Big Bang"  creation of the universe. Same equation, different range of values for the cosmological constant, and different initial conditions.
- In 1929, astronomer Edwin Hubble saw that the universe was expanding. This confirmed Lemaitre's suggestion. At this point, the cosmological constant was assumed to be zero. Same equation, different value for the cosmological constant.
- In the 1970s, Vera Rubin saw that galaxies rotated in a strange way, and this led to the idea of Dark Matter. Same equation, same value for the cosmological constant, but introducing the concept that the mass of the universe is a combination of "normal" matter, and the dominant "Dark" matter.
- In the 1990s, the accelerating expansion of the universe was discovered, attributed to Dark Energy. Same equation, different value for the cosmological constant.

Einstein's field equation has proved remarkably resilient for over a century.
- New theoretical, computational and experimental techniques for investigating it have repeatedly confirmed it to an incredible accuracy
- With applications including black holes, GPS, gravitational waves, gravitational lensing and frame dragging
- Discoveries in this time have changed the interpretation of various parameters in the equation, but the equation itself has stood the test of time.
- Some very recent satellite measurements of the expansion of the universe have called into question just how constant this "cosmological constant" truly is. We will need more data to confirm if it has changed, and if so, in what manner
- An engineering "law": "All constants are variable" (under different conditions)

https://en.wikipedia.org/wiki/Einstein_field_equations#The_cosmological_constant

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General Science / Re: Is There A Pattern In The Number Pi?

« on: 30/09/2024 06:28:47 »

Pi can be represented by a very simple pattern:
π/4 = 1 - 1/3 + 1/5 - 1/7 + etc

This series is very slow to converge, so it's not of any practical use to calculate π

But it does illustrate some important aspects:
- 1/1000001 is zero in the first 6 decimal places
- So any pattern that you might detect in the first 6 decimal places will be destroyed by adding in this factor which does not affect the first 6 decimal places.
- The series continues forever, so it will destroy any other pattern you detect in the decimal expansion

It is necessary that the digits of π (or any transcendental number) do not form a repeating pattern
- But it is not sufficient

If you want to represent π, you could use the infinite series, or adopt the standard mathematical shorthand, and just write "π".

https://www.forbes.com/sites/kevinknudson/2016/04/18/everything-you-wanted-to-know-about-pi-part-4-infinite-series/

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Question of the Week / Re: QotW - 24.09.06 - What is the shutter speed of the eye?

« on: 02/09/2024 10:25:19 »

Focal Length
The human eye has a fixed focal length of around 22mm, which gives a similar field of view to a traditional film camera with a 50mm focal length.

But camera lenses come in a wide variety of focal lengths - and a camera with a zoom lens might have a range of 60:1 in focal length (and corresponding wide range in field of view). 

Saccades
The human eye does not focus on just one image, but darts around parts of the image, allowing the high-resolution macula to dwell on different parts of the image for short periods of time. Saccades are the quick movements from one part of the image to another, occurring around 3-5 times per second, and lasting about 50ms each time. The brain suppresses the blurry image during the saccades, making them effectively unconscious and invisible to the viewer.
- The image collection time on one part of the retina is limited to 200-300ms, which could be considered the maximum "exposure time"
- But the brain is able to build up a detailed image across a number of saccades, forming a composite image like the popular "panorama" photo, or "stacking" in astrophotography.

An interesting comparison here... They suggest that the human eye can't see any more detail after looking at a scene for more than 10 to 15 seconds:
https://www.cambridgeincolour.com/tutorials/cameras-vs-human-eye.htm

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Question of the Week / Re: QotW - 24.09.06 - What is the shutter speed of the eye?

« on: 01/09/2024 11:42:03 »

Some thoughts...

Aperture
The human pupil size varies from 2 mm diameter in bright light to 8 mm under dark conditions. Smartphone camera lenses are in a similar range.
However, consumer hand-held cameras have a light-collecting diameter of around 20mm, and perhaps 100mm with professional telephoto lenses. This means they can collect a lot more photons, under the same conditions.

Fastest Detectable Event
For both cameras and the human eye, extremely short events can be detected if they generate enough photons during the period when the camera aperture (or eyelid) is open. So this is not a characteristic of a camera or the eye.

Minimum Exposure Time
As Alan mentioned, the flicker fusion frequency of the human eye is around 20 frames per second for most people, so you could say that the minimum exposure time for the human eye is around 50ms.
- Some consumer cameras have a minimum exposure time of 1/1000 second, and are able to freeze very fast events which are a blur to the human eye.

Maximum Exposure Time
When making star trails, or using a tracking mount, astrophotography can use exposure times of hours.

The flicker fusion frequency suggests that the maximum exposure time of the human eye is somewhere around 5-10ms in bright conditions.
- Under low-light conditions, the human eye is able to integrate light over a longer period - but I suspect not more than a second. So you could call this the maximum exposure time.

Photon Sensitivity
In modern astronomical CCDs, about 80% of incoming photons striking the sensor will generate a detectable charge in a pixel (the rest are reflected, or strike wiring, etc).

I heard that the human eye requires two photons to strike the same cone cell within a short period to register a flash of light. This will reduce thermal noise, but also reduces sensitivity compared to a CCD camera.

Thermal Noise
Thermal noise generates lots of charges all the time in a CCD camera, so to improve the signal-to-noise ratio, visible-light astronomical telescopes are cooled to liquid nitrogen temperatures.
- Infra-red telescopes like James Webb must be cooled to <7K.
- With a non-cooled sensor, consumer-grade cameras need to collect a lot of photons to ensure that the image stands out above the thermal noise.
- To reduce thermal noise, consumer cameras typically use tricks like taking a reference exposure with the shutter closed, to measure the average thermal noise, and subtract that from the image with the shutter open. This works better with relatively bright objects which generate more than half of the charge of the thermal noise.

Colour Sensitivity
Consumer cameras are able to take colour photos under very low-light conditions, if the exposure time is long enough.

However, the human cone cells which detect colour are only effective under brightly lit conditions; in the dark, human eyesight falls back to the black-and white rod cells, with lower resolution than the cone cells of the central vision.

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Physiology & Medicine / Re: What Is Junk Food?

« on: 27/06/2024 22:20:34 »

Food which is high in fat and sugar has a high calorie content which is inherently appealing to our brain; fast food producers have worked for decades to maximise that appeal, without any concern for making it a balanced meal.
- Fibre is essential to feed our microbiomes and for genera gut health, but this reduces the fat and sugar content, so this has a low priority for junk food manufacturers
- High carbohydrate food sources like potato (French fries), corn (popcorn+butter), wheat (white bread with butter, pastries) and rice (fried rice) are cheap and easy to mass-produce and ship around the country (compared to protein, for example).
- Lots of calories leads to obesity, and easily digested carbohydrates lead to a sugar rush, both of which are risks for insulin resistance and Type 2 diabetes.

It takes more effort to cook vegetables and eat a variety of fresh fruit, and the calorie content is much lower
- Contrast that with squeezed juice which removes the natural fibre and gives as much calories as 5 pieces of fruit
- Or smoothies which contain fat from milk and carbohydrates from fruit

I heard of a food health program for young kids which encouraged them to "eat a rainbow": A variety of coloured fruit and vegetables.

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Question of the Week / Re: QotW - 24.06.28 - Are there stars which aren't part of a galaxy?

« on: 24/06/2024 10:13:59 »

where does the soot come from? (intergalactic burning of fossil fuels no doubt, when will they ever learn)

As Halc commented: Stellar burning of fossil hydrogen...

It is thought that the Big Bang produced about 75% Hydrogen, 25% Helium and a tiny trace of Lithium (plus a heap of photons, neutrinos, and whatever Dark Matter is...).

The early universe was hotter than today, but also denser, so how the hypothetical first "Population III" stars formed is a bit of a mystery. Without heavier elements to catalyse fusion through the CNO cycle, it is thought that they might have been monsters of up to 100 times the mass of the Sun, which went supernova in a million years or so of starting fusion.

Today's "Population I" stars form from dust clouds containing lots of carbon, nitrogen, oxygen, iron and other elements from prior supernovas and heavy elements from neutron star collisions.
- To some, the word "dust" conveys the thought of particles the size of sand.
- To astronomers, the typical particle size in these regions is similar to soot from a smokey candle.
- While visible candle soot is mostly hot carbon compounds, interstellar dust clouds are near absolute zero, and have lots of ices of H₂O, C_yH_x, CO₂, NO_x, NH_x, etc as well as SiO_x, FeO_x, as well as the primordial Hydrogen and Helium.
https://en.wikipedia.org/wiki/Stellar_population#Formation_of_the_first_stars
https://en.wikipedia.org/wiki/Nucleosynthesis

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Question of the Week / Re: QotW - 24.06.28 - Are there stars which aren't part of a galaxy?

« on: 22/06/2024 22:41:13 »

The early development of galaxies (and the supermassive black holes around which they seem to form) is a mystery of modern astronomy.
- Images of active galaxies (Quasars) have been observed surprisingly close to the date of the Big Bang, and astronomers don't understand how they grew so fast
- Astronomers hope that the James Webb Space telescope will shed more light on this, with its infra-red capability to view galaxies with extreme red-shifts

Stars can only form where the density of matter is sufficient to draw matter together, despite the effects of temperature and the motion of atoms to keep it apart
- In today's universe, stars form in Giant Molecular Clouds of gas and soot, at temperatures around 10K (-250C). https://en.wikipedia.org/wiki/Star_formation
- It is not clear how such temperatures would occur in the early universe, which was much hotter than today's 2.7K cosmic microwave background radiation - or near active black holes.
- But it is thought that these densities will occur most often in galaxies or globular clusters.

A recent image by the ESA Euclid telescope shows what appears to be around a trillion stars in intergalactic space, within the Perseus Cluster.
- It is thought that these were flung into space by gravitational interactions with the two massive galaxies in the cluster.

Orphan_Stars_Perseus_Cluster_Euclid.png (1021.35 kB . 1029x708 - viewed 621 times)
https://www.esa.int/Science_Exploration/Space_Science/Euclid/Euclid_s_view_of_the_Perseus_cluster_of_galaxies

A similar gravitational interaction (collision) is expected between our Milky Way galaxy, and the Andromeda galaxy, in about 5 billion years.
- The following simulation shows stars being torn off both galaxies and flung into space as isolated stars (the collision starts at about the 30-second mark).

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General Science / Re: Do germinating tomatoes kill off competing plants?

« on: 12/06/2024 10:26:46 »

In the real world, once the initial shoot reaches the surface of the soil, it can spread, and shade out all of the lower shoots.
- This video is a bit artificial, in that the underground shoots manage to get some light through the sides of the glass container...

The first shoot to reach the air can transpire, and suck up nutrients from the soil. The other underground shoots can't transpire, and so have a much lower supply of water and nutrients.

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Physics, Astronomy & Cosmology / Re: Is there any plausible mechanism for solar flares affecting health?

« on: 21/02/2024 09:18:10 »

It is known that some birds navigate via magnetic fields (and most birds fly not much higher than ground level).
- So some birds could get lost (or perhaps suffer vertigo) from changing geomagnetic fields.

During the Carrington event in 1859, currents induced in long-distance telegraph lines reportedly caused electrical shocks to telegraph operators, and caused some fires.
https://en.wikipedia.org/wiki/Carrington_Event#Telegraphs

Today we are far more dependent on long-distance conductors than in the 1850s - for electrical power transmission, water, oil and gas pipelines. A small flare in March 1989 took out mains power in Quebec for 10 hours; there could be severe loss of life if there were an extended power and communications outage during the depths of winter.
https://en.wikipedia.org/wiki/March_1989_geomagnetic_storm#Quebec_power_blackout

It appears that Schumann resonance is a real thing, and is continually randomly triggered by random lightning strikes around the world, with a fundamental frequency around 7.8Hz, with harmonics extending up to 34Hz.
https://en.wikipedia.org/wiki/Schumann_resonances#Basic_theory

It is conceivable that this Schumann resonance could be triggered in a coherent way by geomagnetic storms (as distinct from random excitation from lightning), but the main frequencies of geomagnetic storms are around 0.0001 Hz to 0.004 Hz (ie the milliHertz band). The wavelength of these disturbances is far too long to resonate in the cavity of the Earth's ionosphere. 
https://ieeexplore.ieee.org/document/9794666

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Plant Sciences, Zoology & Evolution / Re: Can climate change cause queen bees & wasps to come out of hibernation too early

« on: 09/02/2024 20:07:07 »

There is a risk during climate change that the lifecycles of animals and their food sources will become misaligned, with adverse consequences.
- Bees that come out of overwintering before (or after) the flowers bloom would be bad for both flowers and bees
- Birds that hatch before (or after) the insects take flight would be bad for both birds (and us)

Nature can adapt to slow changes in the environment; the too-early bird and the too-late bird won't catch as many worms as the just-in-time bird. Presumably the just-in-time bird would have more chicks, and over time, the genetic mix of the population will shift towards the optimum.
- The concern is whether climate changes are happening faster than the population's genetic adaptation
- Or whether the one additional stress of climate change will add to other stressors like habitat loss, chemical pollution, etc to cause additional extinctions.

Since I live in a non-snowy environment, bees are unlikely to take extreme measures like those in an environment where all plants are covered by a thick layer of snow in winter.

https://beeswiki.com/do-bees-hibernate/

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Guest Book / Re: Why Are There So Few People On Thenakedscientist?

« on: 07/02/2024 21:10:04 »

Probably Google is still viewing the questions and answers.

But for short answers, Google seems to prefer sources like Stack Exchange, which probably has a lot more live discussion...

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Physics, Astronomy & Cosmology / Re: If Andromeda is red shifted how can we collide with it?

« on: 07/02/2024 21:04:57 »

A.I. systems are stochastically good....All it had to do was to choose between Red or Blue

If you look at the probability of finding "red shift" or "blue shift" in published literature, there will be a heavy preponderance of "red shift". So statistically, an AI is going to say "red shift" nearly all the time.
- If the topic is narrowed to Andromeda (one of the few cases where red shift does not apply), you might end up selecting one of the niche cases in the literature where red shift is paired with "negative".

Contrast this with "Doppler Shift", where we are familiar with ambulance sounds, radar returns, ultrasound echograms and laser speed traps measuring both approaching and receding objects. Doppler Shift has much less of an inference that objects are receding, but it's mostly applied to terrestrial objects (and sometimes space probes).

When you talk astronomy, "red shift" is predominant.

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Physics, Astronomy & Cosmology / Re: If Andromeda is red shifted how can we collide with it?

« on: 05/02/2024 22:09:09 »

Velocity has a direction (when measured against some coordinate system); negative velocity is in the opposite direction...

Red-Shift has a direction (changing emission lines towards longer wavelengths); 0 red shift means "no relative velocity"; conceivably a blue shift (changing emission lines towards shorter wavelengths) would be a negative red-shift. ie the concept appears well-defined, even if it is not conventional way of expressing it...

Note that the meaning of "zero red-shift" has been changing over the years; modern spectrometers used for discovering exoplanets can measure the red-shift of star with a relative velocity of a walking pace (superimposed on Earth's orbit around the Sun of around 30 km/second).

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Physics, Astronomy & Cosmology / Re: If Andromeda is red shifted how can we collide with it?

« on: 31/01/2024 09:12:17 »

This article suggests that there are around 100 known blue-shifted galaxies (of which Andromeda is one).
https://en.wikipedia.org/wiki/Andromeda_Galaxy#Collision_with_the_Milky_Way

While we can measure the radial velocity of Andromeda quite accurately by its blue-shift, there was greater uncertainty about the "sideways" motion of Andromeda, and indeed whether Andromeda would hit the Milky Way head on, or miss altogether.
- The tangential velocity estimate was improved by using images taken by Hubble (the satellite) a decade apart, and comparing the position of stars in Andromeda against the position of distant galaxies.
- More recently, the tangential velocity estimate was improved again by using observations by ESA's Gaia satellite, which was able to take precise measurements of the positions of individual bright stars in Andromeda. This confirms that the collision will occur,  but later, after an initial glancing blow.
https://www.esa.int/Science_Exploration/Space_Science/Gaia/Gaia_clocks_new_speeds_for_Milky_Way-Andromeda_collision

Edwin Hubble (the person) discovered that most galaxies are red-shifted; this applies to distant galaxies, where random local motions are dominated by the general expansion of the universe.

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Physics, Astronomy & Cosmology / Re: How many stars in the most dark sky without {The_Sun-0001} naked eye with 20/20!

« on: 25/01/2024 21:02:20 »

An answer I recall from long ago was around 7,000 stars brighter than magnitude 6.
- Looking on the internet, I see answers between 2,000 and 10,000 (the latter if you were to roam to both hemispheres).

Neil deGrasse Tyson grew up in the canyons of New York CIty, and tells of the time he first went to the Haydn planetarium.
- When they threw up an image of thousands of stars, he thought it was a joke. He had only seen about 4 at a time...
- He is now director of the Haydn planetarium

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Physics, Astronomy & Cosmology / Re: What happens to a pair of entangled quanta if one of them fell in a Black Hole ?

« on: 17/01/2024 09:25:23 »

if one (entangled) particle is destroyed the other can happily enjoy a solo career ?

Having entangled particles is not rare - it is the normal situation for most things in the universe.
- Every time a particle bumps into another (directly or indirectly), their quantum states become entwined/entangled.
- Pretty soon, every particle on Earth is entangled with every other particle on Earth.
- It takes longer, but particles on Earth can become entangled by photons from the star Sirius.

It is the exception to have a particle which is in a known quantum state - it takes a lot of work to get it into a known state, and it is easily disrupted by coming into contact with light, heat or sound in the environment, effectively entangling it once again with the unknown quantum state of "the universe".

So an entangled particle falling into a black hole is quite normal, and the rest of the surrounding universe continues unaffected - although the plasma in the accretion disk may not enjoy this liberty for very much longer...

So don't ewe stray beyond the horizon of your paddock; in that event you may become entangled with the fence (and by extension, the whole Earth).

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Physiology & Medicine / Re: Are Right Handed People Right Footed ?

« on: 17/01/2024 09:07:11 »

Ultimately, hand/hoof preference comes down to a brain asymmetry.

Whether the hand/hoof preference created the brain asymmetry or vice-versa is probably hard to pin down.

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Physics, Astronomy & Cosmology / Re: What happens to a pair of entangled quanta if one of them fell in a Black Hole ?

« on: 16/01/2024 10:10:23 »

There is a rule that you can't clone a quantum state - that is equivalent to measuring it, which changes its state.

However, since a particle in the black hole can't directly affect anything outside the event horizon, there is a theoretical possibility that the quantum state remains on the event horizon, as well as falling in.
https://en.wikipedia.org/wiki/No-cloning_theorem

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New Theories / Re: New Maths? (Orders of Magnitude)

« on: 14/01/2024 21:01:18 »

Once you have a basic idea of orders of magnitude, you can view the universe at many different scales, introducing ideas of maps, the world, astronomy, as well as biology, chemistry and nuclear physics...

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New Theories / New Maths? (Orders of Magnitude)

« on: 14/01/2024 11:07:59 »

An idea for a way of teaching maths that is better related to human senses. This was split off from:
https://www.thenakedscientists.com/forum/index.php?topic=86686.msg719574#msg719574
All comments are welcome! Feel free to pass these ramblings on to real educators!

Human Senses
Most human senses need to span many orders of magnitude in sensitivity, so the response tends to be logarithmic, rather than linear (linear would imply 50% is half-way).

I think that kids would find maths and science more relatable if we talked in units they can feel and relate to - logarithmic units
-   Sound intensity is measured in dB (logarithmic), sound frequency is measured in octaves (logarithmic), brightness in astronomy is measured in magnitudes (logarithmic)
-   Bicycle gears form a geometric progression
-   Hardness is measured on Moh's scale (logarithmic except for diamond)
-   Earthquake strength uses the Richter scale, and wind speed uses the Beaufort scale (both logarithmic)
-   Electronic component values fall on a logarithmic scale
- I suspect that things like touch, smell and taste would also have a logarithmic response
- Psychologists have even shown that abstract concepts like "value" or "wealth" also have a logarithmic response
- The Metric System and Scientific notation are both based on a logarithmic scale, with units like meters and grams, and their derived units of millimeters, kilometers, kilograms and tonnes.

Number Concepts
Subitizing: I think that a natural place to start with numbers is with subitizing, the innate ability to count 0 to 4 or 5 objects at a glance. Learning to relate the name and visual representation of the digit to the quantity of objects is important (and 0 is vital, as the thing kids feel most strongly is if someone else gets 1 lolly, and they get 0).

Large Orders of Magnitude: 0, 1, 10, 100, 1000: When I was a kid, we were given number blocks, where "1" was represented by a small white cube.
-   10 of these in a line represents 1 order of magnitude (and 1 dimension)
-   100 of these in a 10x10 plate represents 2 orders of magnitude (and 2 dimensions)
-   1000 of these in a 10x10x10 cube represents 3 orders of magnitude (and 3 dimensions)

This should be supplemented by kinesthetic experiences like:
-   Walk a metre: A kilometre is 1000 times farther
-   Walk 10m: A kilometre is 100 times farther
-   Walk 100m: A kilometre is 10 times farther
-   Walk 1 km
-   And similar sensory experiences with mass, brightness, sound intensity, frequency, and smell (taste might be a bit more of risk due to potential allergies?)
- Wealth comparisons like kids who survive on $1/day, $10/day, $100/day and $1000/day
-   Given random quantities of each, which is it closer to? (estimating orders of magnitude, bearing in mind that 3 is closer to mid-way on a logarithmic scale)

Small Orders of Magnitude: 1, 0.1, 0.01, 0.001, 0: Here you could start with the large cube, and the plate and line represent fractions of the whole (whole apple, whole bottle of water, etc)
This should be supplemented by kinesthetic experiences like:
-   Take a metre of string, compare to 10cm, 1com and 1mm of string
-   Take a kilo of sand, compare to 100g, 10g and 1g
-   Take a litre of water, compare to 100ml, 10ml, 1ml
-   Compare a walk of 1km to 100m, 10m, 1m
-   Compare $10, $1, 10c, 1c
-   Given random quantities of each, which is it closer to? (estimating orders of magnitude, bearing in mind that 3 is closer to mid-way on a logarithmic scale)
-   And similar sensory experiences with brightness, sound intensity, frequency, and smell
-   Not forgetting 0 of each, which is a child's ultimate reference point! (and the control value in an experiment, eg no sound, no light, no money)

Simple Arithmetic and counting further: Based on Subitization, what happens if you add one, or subtract one, take all away or add another group the same size?
-   When adding 4+4, you quickly run into answers that are beyond subitizing
-   This should lead to learning numbers from 0 to 10
-   And the more laborious process of counting with 1-to-1 correspondence (which it takes kids a while to achieve)

The leading digits: After the orders of magnitude, the next most important numbers are:
-   Nx 3 (the sensory middle of 1 & 10), 30, 300 [Mathematically 10^1/₂  = 3.16, but use 3 as an integer]
-   Nx 2 & 5 (break the decade into 3 ranges), 20, 50, 200, 500 [10^1/₃=2.15 & 10^2/₃=4.64, but use 2 & 5 as integers]
-   Doubling 1, 2, 4, 8 (especially important in music and computers)
-   N x 9, the last digit to be covered (and least useful, being almost imperceptibly different from 10), 90, 900. Immunise against the shopper's fallacy: The gap between 9 & 10 is much smaller than between 1 & 2!

Next most important numbers: According to Benford's law, when looking at numbers in physical laws, finance and election results, about 30% of the time the leading digit will be "1" (which is why we start with the orders of magnitudes).
-   This means that the next most important numbers are between 11 & 19
-   Then fill in the gaps up to 100.
-Give the ideas of units, 10s, 100s in position notation (and digit 0)

Simple Scientific notation: Count how many orders of magnitude larger or smaller two quantities are.
Have blocks that increase in powers of 2:
-   Unit block, line of 2, plate of 4, cube of 8 = 3 dimensions
-   cube of 8, line of 2 cubes of 8, plate of 4 cubes of 8, cube of 8 cubes of 8 = 64
-   Count the powers of 2

Logic: AND, OR,NOT with computer simulation of real examples.

Fractions: Teach it with music, pies, and apples:
-   Octave 2:1; harmonies 2/3, 3/4, etc
-   For lengths, mass, volume, etc, start with 1/10 and progress to other ratios like ¹/₂, ¹/₃, ²/₃,¹/₄, ²/₄=¹/₂,³/₄, etc

Tools Required
-   Weights and liquid volumes can be easily obtained
-   Computers are easily able to present random groupings of dots, apples, oranges and coins?
-   1-1 correspondence could be helped with touch screens which make the object visibly different after you touch it and count it
-   Sound levels and sound frequencies over several orders of magnitude are easily generated on modern computers
-   Various strengths of scents tend to be single-use, but I am sure they could be produced faily easily. Ensure you go from weakest to strongest, otherwise the nose will become desensitised!
-   Computers are able to simulate simple logical combinations, but they would need to be child-relatable examples, eg assume pet = cat or dog; so an elephant is not a pet?

Potential Difficulties?
It's different! The idea of teaching 10, 100 & 1000 before teaching 9 seems radical.
Logarithmic units might be a difficult concept for existing primary school teachers to grasp

Several examples were kindly contributed by alancalverd & vhfpmr