In other languages: Deutsch Italiano 日本語 한국어 Polski Português, Brasil Русский Українська 简体中文

Technologies: Difference between revisions

From Official Factorio Wiki
Jump to navigation Jump to search
(Fix artillery shooting speed cost as well (×3 then - 2000 is the nicest way I could figure out for how to write it))
 
(15 intermediate revisions by 10 users not shown)
Line 1: Line 1:
{{Languages}}
{{Languages}}
[[File:Technology tree.png|thumb|right|280px|The technology tree − Click to enlarge.]]
[[File:tech_bonuses.png|thumb|right|280px|Researched tech bonuses.]]
'''Technologies''' are [[Research|researched]] in [[lab]]s using [[science pack]]s.


Technologies are [[Research|researched]] in [[lab]]s using [[science pack]]s.
Every technology needs a certain amount of science packs per science pack type to be researched, for example the [[Logistics (research)|Logistics]] technology requires 20 [[automation science pack]]s to be placed in labs while the technology is selected for research.
 
Every technology needs a certain amount of science packs per science pack type to be researched, for example the [[logistics 1]] technology requires 20 [[science pack 1]] to be placed in labs while the technology is selected for research.


== Achievements ==
== Achievements ==
{{Main|Achievements}}
{{Achievement|tech-maniac}}
Technologies are directly connected to the following achievement:
* Completing infinite technologies of any level is not required for '''Tech maniac'''. All non-infinite levels of technologies that have infinite continuations are still required.
; Tech maniac
{{clear}}
: Research all technologies.  (0.15 - This requires at least one rank of each technology, including the infinite technologies.  See [https://forums.factorio.com/viewtopic.php?f=23&t=45091 Forum post]. )
== Infinite technologies ==


== All technologies ==
While most technologies in Factorio are either one-off or have a finite, relatively small number of levels available, a few are "infinite", meaning the player can research as many levels as they can afford. All of them unlock bonuses to existing technologies, never new structures or abilities. The per-level bonuses are constant for a particular infinite technologies and, like finite research bonuses, are additive within a single technology. They are subject to diminishing returns; thus, the per-level and per-science pack contributions from very high levels of infinite technologies will eventually provide only marginal improvements.
This is a list of all technologies that are currently researchable in the game.


The tree of technologies:
All infinite technologies levels require [[space science pack]]s, and are also the only technologies that do. As such, they are late-game technologies intended primarily for players who wish to continue playing and expand their factory past the nominal victory condition of launching a rocket.


[[File:Technology tree.png|800px]]
Infinite technologies are identified in-game by a small <code>∞</code> infinity symbol shown in the top right corner of the research technology's card in the research screen.


=== Machines ===
Most infinite technologies are continuations of ordinary multi-level technologies; the "infinite" mechanic becomes effective once the player reaches the card initially labeled with <code>N - ∞</code> in the research tree. Only the two [[artillery]]-related technologies (artillery shell [[artillery shell range (research)|range]] and [[artillery shell shooting speed (research)|shooting speed]]) are infinite-only; for these, <code>1 - ∞</code> is shown before any levels in them are researched. In either case, once the first infinite level is researched, the card label switches to the one discussed above.


* [[Advanced electronics (research)|Advanced electronics]]
=== Pricing formulas ===
** [[Advanced electronics 2 (research)|Advanced electronics 2]]
* [[Automation (research)|Automation]]
** [[Automation 2 (research)|Automation 2]]
** [[Automation 3 (research)|Automation 3]]
* [[Effect transmission (research)|Effect transmission]]
* [[Electric energy accumulators (research)|Electric energy accumulators]]
* [[Electric energy distribution 1 (research)|Electric energy distribution 1]]
** [[Electric energy distribution 2 (research)|Electric energy distribution 2]]
* [[Fluid handling (research)|Fluid handling]]
* [[Logistics (research)|Logistics]]
** [[Logistics 2 (research)|Logistics 2]]
** [[Logistics 3 (research)|Logistics 3]]
*[[Oil processing (research)|Oil processing]]
*[[Rocket silo (research)|Rocket silo]]
*[[Solar energy (research)|Solar energy]]
*[[Stack inserter (research)|Stack inserter]]


=== [[Military]] ===
The price of all infinite technologies is generated in a [[Wikipedia:Progression|mathematical progression]]; for the majority of technologies, the progression is [[Wikipedia:Geometric_progression|geometric]], mostly in powers of 2. Two technologies - [[mining productivity (research)|mining productivity]] and [[follower robot count (research)|follower robot count]] - use an [[Wikipedia:Arithmetic_progression|arithmetic progression]] instead.


* [[Cluster grenade (research)|Cluster grenade]]
The table below summarizes for all infinite researches their first infinite level, the cost of the first few infinite levels, the cost formula and the per-level bonus.
* [[Combat robotics (research)|Combat robotics]]
** [[Combat robotics 2 (research)|Combat robotics 2]]
** [[Combat robotics 3 (research)|Combat robotics 3]]
* [[Flamethrower (research)|Flamethrower]]
* [[Military (research)|Military]]
** [[Military 2 (research)|Military 2]]
** [[Military 3 (research)|Military 3]]
** [[Military 4 (research)|Military 4]]
* [[Rocketry (research)|Rocketry]]
** [[Explosive rocketry (research)|Explosive rocketry]]


=== Bonuses ===
We denote by <code>N</code> the current level of the research, by <code>F</code> the final non-infinite level of the research (hence F+1 is the first "infinite" level) and by <code>P[N]</code> the price of the research at level <code>N</code>.
{|class="wikitable"
! Technology !! Bonus !! Science Packs !! F+1 !! P[N] !! P[F+1] !! P[F+2] !! P[F+3] !! P[F+4] !! P[F+5] !! ...
|-
| {{Icontech|Worker robot speed (research)}} [[Worker robot speed (research)|Worker robot speed ]] || +65% Robot Speed || {{icon|Automation science pack}} {{icon|Logistic science pack}} {{icon|Chemical science pack}} {{icon|Production science pack}} {{icon|Utility science pack}} {{icon|Space science pack}} || style=text-align:center|7 || 1,000 × 2^(N - F - 1) || style=text-align:center|1,000 || style=text-align:center|2,000 || style=text-align:center|4,000 || style=text-align:center|8,000 || style=text-align:center|16,000 || style=text-align:center|×2
|-
| {{Icontech|Mining productivity (research)}} [[Mining productivity (research)|Mining productivity]] || +10% Mining Productivity || {{icon|Automation science pack}} {{icon|Logistic science pack}} {{icon|Chemical science pack}} {{icon|Production science pack}} {{icon|Utility science pack}} {{icon|Space science pack}} || style=text-align:center|4 || 2,500 × (N - F) || style=text-align:center|2,500 || style=text-align:center|5,000 || style=text-align:center|7,500 || style=text-align:center|10,000 || style=text-align:center|12,500 || style=text-align:center|+2500
|-
| {{Icontech|Physical projectile damage (research)}} [[Physical projectile damage (research)|Physical projectile damage]] || +40% Bullet Damage <br> +70% Turret Damage <br> +40% Shotgun Shell Damage <br> +100% Cannon Shell Damage || {{icon|Automation science pack}} {{icon|Logistic science pack}} {{icon|Chemical science pack}} {{icon|Military science pack}} {{icon|Utility science pack}} {{icon|Space science pack}} || style=text-align:center|7 || 1,000 × 2^(N - F - 1) || style=text-align:center|1,000 || style=text-align:center|2,000 || style=text-align:center|4,000 || style=text-align:center|8,000 || style=text-align:center|16,000 || style=text-align:center|×2
|-
| {{Icontech|Stronger explosives (research)}} [[Stronger explosives (research)|Stronger explosives]] || +50% Rocket Damage <br> +20% Grenade Damage <br> +20% Landmine Damage || {{icon|Automation science pack}} {{icon|Logistic science pack}} {{icon|Chemical science pack}} {{icon|Military science pack}} {{icon|Utility science pack}} {{icon|Space science pack}} || style=text-align:center|7 || 1,000 × 2^(N - F - 1) || style=text-align:center|1,000 || style=text-align:center|2,000 || style=text-align:center|4,000 || style=text-align:center|8,000 || style=text-align:center|16,000 || style=text-align:center|×2
|-
| {{Icontech|Refined flammables (research)}} [[Refined flammables (research)|Refined flammables]] || +20% Flame Thrower Damage {{icon|Flamethrower ammo}} {{icon|Flamethrower turret}}|| {{icon|Automation science pack}} {{icon|Logistic science pack}} {{icon|Chemical science pack}} {{icon|Military science pack}} {{icon|Utility science pack}} {{icon|Space science pack}} || style=text-align:center|7 || 1,000 × 2^(N - F - 1) || style=text-align:center|1,000 || style=text-align:center|2,000 || style=text-align:center|4,000 || style=text-align:center|8,000 || style=text-align:center|16,000 || style=text-align:center|×2
|-
| {{Icontech|Energy weapons damage (research)}} [[Energy weapons damage (research)|Energy weapons damage]] || +70% Laser Damage {{icon|Laser turret}} {{icon|Personal laser defense}} {{icon|Distractor capsule}}<br> +70% Electric Damage {{icon|Discharge defense}}<br> +30% Beam Damage {{icon|Destroyer capsule}} || {{icon|Automation science pack}} {{icon|Logistic science pack}} {{icon|Chemical science pack}} {{icon|Military science pack}} {{icon|Utility science pack}} {{icon|Space science pack}} || style=text-align:center|7 || 1,000 × 2^(N - F - 1) || style=text-align:center|1,000 || style=text-align:center|2,000 || style=text-align:center|4,000 || style=text-align:center|8,000 || style=text-align:center|16,000 || style=text-align:center|×2
|-
| {{Icontech|Artillery shell range (research)}} [[Artillery shell range (research)|Artillery shell range]] || +30% Artillery Shell Range || {{icon|Automation science pack}} {{icon|Logistic science pack}} {{icon|Chemical science pack}} {{icon|Military science pack}} {{icon|Utility science pack}} {{icon|Space science pack}} || style=text-align:center|1 || 1,000 × 2^(N - F) || style=text-align:center|2,000 || style=text-align:center|4,000 || style=text-align:center|8,000 || style=text-align:center|16,000 || style=text-align:center|32,000 || style=text-align:center|×2
|-
| {{Icontech|Artillery shell shooting speed (research)}} [[Artillery shell shooting speed (research)|Artillery shell speed]] || +100% Artillery Speed || {{icon|Automation science pack}} {{icon|Logistic science pack}} {{icon|Chemical science pack}} {{icon|Military science pack}} {{icon|Utility science pack}} {{icon|Space science pack}} || style=text-align:center|1 || 1,000 + 1,000 × 3^(N - F - 1) || style=text-align:center|2,000 || style=text-align:center|4,000 || style=text-align:center|10,000 || style=text-align:center|28,000 || style=text-align:center|82,000 || style=text-align:center|×3 then - 2000
|-
| {{Icontech|Follower robot count (research)}} [[Follower robot count (research)|Follower robot count]] || +10 Follower Robots || {{icon|Automation science pack}} {{icon|Logistic science pack}} {{icon|Chemical science pack}} {{icon|Military science pack}} {{icon|Production science pack}} {{icon|Utility science pack}} {{icon|Space science pack}} || style=text-align:center|7 || 100 × (N - F) + 900 || style=text-align:center|1,000 || style=text-align:center|1,100 || style=text-align:center|1,200 || style=text-align:center|1,300 || style=text-align:center|1,400 || style=text-align:center|+100
|}


* [[Bullet damage (research)|Bullet damage]]
=== Cumulative cost ===
* [[Bullet shooting speed (research)|Bullet shooting speed]]
* [[Combat robot damage (research)|Combat robot damage]]
* [[Flamethrower damage (research)|Flamethrower damage]]
* [[Follower robot count (research)|Follower robot count]]
* [[Grenade damage (research)|Grenade damage]]
* [[Gun turret damage (research)|Gun turret damage]]
* [[Inserter capacity bonus (research)|Inserter capacity bonus]]
* [[Lab research speed (research)|Lab research speed]]
* [[Laser turret damage (research)|Laser turret damage]]
* [[Laser turret shooting speed (research)|Laser turret shooting speed]]
* [[Rocket damage (research)|Rocket damage]]
* [[Rocket shooting speed (research)|Rocket shooting speed]]
* [[Shotgun shell damage (research)|Shotgun shell damage]]
* [[Shotgun shell shooting speed (research)|Shotgun shell shooting speed]]
* [[Worker robot cargo size (research)|Worker robot cargo size]]
* [[Worker robot speed (research)|Worker robot speed]]


=== Player augmentation ===
As the price of most infinite technologies (specifically, those based on geometric progressions) increases very steeply, it may be a good idea for players to set realistic target levels for each of the infinite technologies they wish to pursue, and make their factory plans accordingly. To that end, the following properties of cumulative infinite research prices may be useful:


* [[Automated construction (research)|Automated construction]]
#For infinite technologies whose underlying equation is a powers-of-two geometric series, the cumulative price of the first <code>N - F</code> infinite levels (skipping the first <code>F</code> level, so counting "infinite" levels only) is <code>2 × P[N] - P[F+1]</code>; i.e., twice the price of the final researched level, less the price of the first "infinite" level.
* [[Character logistic slots (research)|Character logistic slots]]
#*As ''N'' increases, this is approximated well by <code>2 × P[N] = P[N+1]</code>, so the cumulative cost of researching to level <code>N</code> is about as much as researching level <code>N+1</code>.
** [[Character logistic trash slots (research)|Character logistic trash slots]]
#*If one decides a level <code>M</code> which one considers the "highest feasible" with their current science pack production capacity, expanding said capacity by a factor of <code>X</code> will allow about <code>log[2](X)</code> additional levels to be researched before the next level takes longer to research with the expanded capacity than level <code>M + 1</code> would have taken with the pre-expansion production capacity.
** [[Auto character logistic trash slots (research)|Auto character logistic trash slots]]
#*For example, if one expands production capacity by a factor of 10, they will be able to research at least <code>floor(log[2](10)) = 3</code> and at most <code>ceiling(log[2](10)) = 4</code> additional levels in a given technology before the exponential increase in price negates the speed benefits of their ×10 capacity expansion.
* [[Construction robotics (research)|Construction robotics]]
#The cumulative price of the first <code>N - F</code> levels of infinite technologies whose underlying equation is an arithmetic series is <code>(N - F) × (P[N] + P[F + 1]) ÷ 2</code>; i.e, <code>N - F</code> times the mean of the prices of the first and last "infinite" level. For the [[Follower robot count (research)]], an additional <code>900 × (N-F)</code> need to be added.
* [[Logistic robotics (research)|Logistic robotics]]
#*Expanding production capacity by a factor of <code>X</code>, as above, will in this case allow an additional <code>N × (X - 1)</code> levels to be researched before the benefit of the expansion is wiped out (i.e., research progress speed drops to or below what it was pre-expansion).
* [[Toolbelt (research)|Toolbelt]]
#The cumulative price of the first '''''N''''' levels of '''artillery shell shooting speed''', the sole infinite technology whose underlying equation is a powers-of-three geometric series (equation type (2)) is <code>1.5 × P[N] - 0.5 × P[1]</code>; i.e., 1.5 times the price of the final researched level, less half the price of the first level.


=== [[Defense]] ===
Note that these prices reflect '''research units''', which will not be equal to science packs if [[productivity module|productivity modules]] are used in labs. (In that case, the science pack requirement will be lower.)
 
* [[Armor crafting (research)|Armor crafting]]
** [[Armor crafting 2 (research)|Armor crafting 2]]
** [[Modular armor (research)|Modular armor]]
* [[Gates (research)|Gates]]
* [[Landmines (research)|Landmines]]
* [[Power armor (research)|Power armor]]
** [[Power armor 2 (research)|Power armor 2]]
* [[Stone walls (research)|Stone walls]]
* [[Turrets (research)|Gun turrets]]
** [[Laser turrets (research)|Laser turrets]]
 
=== Crafting ===
 
* [[Advanced material processing (research)|Advanced material processing]]
** [[Advanced material processing 2 (research)|Advanced material processing 2]]
* [[Advanced oil processing (research)|Advanced oil processing]]
* [[Alien technology (research)|Alien technology]]
* [[Battery (research)|Battery]]
* [[Concrete (research)|Concrete]]
* [[Electronics (research)|Electronics]]
* [[Engine (research)|Engine]]
** [[Electric engine (research)|Electric Engine]]
* [[Explosives (research)|Explosives]]
* [[Flammables (research)|Flammables]]
* [[Flight (research)|Flight]]
* [[Landfill (research)|Landfill]]
* [[Laser (research)|Laser]]
* [[Logistic system (research)|Logistic system]]
* [[Optics (research)|Optics]]
* [[Plastics (research)|Plastics]]
* [[Robotics (research)|Robotics]]
* [[Steel processing (research)|Steel processing]]
* [[Sulfur processing (research)|Sulfur processing]]
 
=== Transportation ===
 
* [[Automated rail transportation (research)|Automated rail transportation]]
* [[Automobilism (research)|Automobilism]]
* [[Rail signals (research)|Rail signals]]
* [[Railway (research)|Railway]]
* [[Tanks (research)|Tanks]]
 
=== [[Armor modules]] ===
 
* [[Battery equipment (research)|Battery equipment]]
** [[Battery MK2 equipment (research)|Battery equipment 2]]
* [[Discharge defense (research)|Discharge defense]]
* [[Energy shield equipment (research)|Energy shield equipment]]
** [[Energy shield  MK2 equipment (research)|Energy shield equipment 2]]
* [[Exoskeleton equipment (research)|Exoskeleton equipment]]
* [[Nightvision equipment (research)|Nightvision equipment]]
* [[Personal laser defense (research)|Personal laser defense]]
* [[Portable fusion reactor (research)|Portable fusion reactor]]
* [[Portable solar panel (research)|Portable solar panel]]
* [[Personal roboport (research)|Personal roboport]]
** [[Personal roboport 2 (research)|Personal roboport 2]]
 
=== [[Module|Modules]] ===
 
* [[Modules (research)|Modules]]
* [[Efficiency module (research)|Efficiency module]]
* [[Productivity module (research)|Productivity module]]
* [[Speed module (research)|Speed module]]
 
== Obsolete technologies ==
This is a list of obsolete technologies that were removed.
{|
| {{icontech|Advanced-chemistry-research||Advanced chemistry}} [[Advanced chemistry (research)|Advanced chemistry]]
| {{icontech|Chemistry-research||Chemistry}} [[Chemistry (research)|Chemistry]]
| {{icontech|iron-working||Iron working}} [[Iron working]]
| {{icontech|rocket-defense||Rocket defense (research)}} [[Rocket defense (research)|Rocket defense]]
|-
|}


== See also ==
== See also ==
Line 163: Line 68:
* [[Science pack]]
* [[Science pack]]


 
{{TechNav}}
[[Category: Technology]]

Latest revision as of 13:30, 29 May 2023

The technology tree − Click to enlarge.
Researched tech bonuses.

Technologies are researched in labs using science packs.

Every technology needs a certain amount of science packs per science pack type to be researched, for example the Logistics technology requires 20 automation science packs to be placed in labs while the technology is selected for research.

Achievements

Tech-maniac-achievement.png Tech maniac

Research all technologies.

  • Completing infinite technologies of any level is not required for Tech maniac. All non-infinite levels of technologies that have infinite continuations are still required.

Infinite technologies

While most technologies in Factorio are either one-off or have a finite, relatively small number of levels available, a few are "infinite", meaning the player can research as many levels as they can afford. All of them unlock bonuses to existing technologies, never new structures or abilities. The per-level bonuses are constant for a particular infinite technologies and, like finite research bonuses, are additive within a single technology. They are subject to diminishing returns; thus, the per-level and per-science pack contributions from very high levels of infinite technologies will eventually provide only marginal improvements.

All infinite technologies levels require space science packs, and are also the only technologies that do. As such, they are late-game technologies intended primarily for players who wish to continue playing and expand their factory past the nominal victory condition of launching a rocket.

Infinite technologies are identified in-game by a small infinity symbol shown in the top right corner of the research technology's card in the research screen.

Most infinite technologies are continuations of ordinary multi-level technologies; the "infinite" mechanic becomes effective once the player reaches the card initially labeled with N - ∞ in the research tree. Only the two artillery-related technologies (artillery shell range and shooting speed) are infinite-only; for these, 1 - ∞ is shown before any levels in them are researched. In either case, once the first infinite level is researched, the card label switches to the one discussed above.

Pricing formulas

The price of all infinite technologies is generated in a mathematical progression; for the majority of technologies, the progression is geometric, mostly in powers of 2. Two technologies - mining productivity and follower robot count - use an arithmetic progression instead.

The table below summarizes for all infinite researches their first infinite level, the cost of the first few infinite levels, the cost formula and the per-level bonus.

We denote by N the current level of the research, by F the final non-infinite level of the research (hence F+1 is the first "infinite" level) and by P[N] the price of the research at level N.

Technology Bonus Science Packs F+1 P[N] P[F+1] P[F+2] P[F+3] P[F+4] P[F+5] ...
Worker robot speed (research).png
Worker robot speed
+65% Robot Speed
Automation science pack.png
Logistic science pack.png
Chemical science pack.png
Production science pack.png
Utility science pack.png
Space science pack.png
7 1,000 × 2^(N - F - 1) 1,000 2,000 4,000 8,000 16,000 ×2
Mining productivity (research).png
Mining productivity
+10% Mining Productivity
Automation science pack.png
Logistic science pack.png
Chemical science pack.png
Production science pack.png
Utility science pack.png
Space science pack.png
4 2,500 × (N - F) 2,500 5,000 7,500 10,000 12,500 +2500
Physical projectile damage (research).png
Physical projectile damage
+40% Bullet Damage
+70% Turret Damage
+40% Shotgun Shell Damage
+100% Cannon Shell Damage
Automation science pack.png
Logistic science pack.png
Chemical science pack.png
Military science pack.png
Utility science pack.png
Space science pack.png
7 1,000 × 2^(N - F - 1) 1,000 2,000 4,000 8,000 16,000 ×2
Stronger explosives (research).png
Stronger explosives
+50% Rocket Damage
+20% Grenade Damage
+20% Landmine Damage
Automation science pack.png
Logistic science pack.png
Chemical science pack.png
Military science pack.png
Utility science pack.png
Space science pack.png
7 1,000 × 2^(N - F - 1) 1,000 2,000 4,000 8,000 16,000 ×2
Refined flammables (research).png
Refined flammables
+20% Flame Thrower Damage
Flamethrower ammo.png
Flamethrower turret.png
Automation science pack.png
Logistic science pack.png
Chemical science pack.png
Military science pack.png
Utility science pack.png
Space science pack.png
7 1,000 × 2^(N - F - 1) 1,000 2,000 4,000 8,000 16,000 ×2
Energy weapons damage (research).png
Energy weapons damage
+70% Laser Damage
Laser turret.png
Personal laser defense.png
Distractor capsule.png

+70% Electric Damage
Discharge defense.png

+30% Beam Damage
Destroyer capsule.png
Automation science pack.png
Logistic science pack.png
Chemical science pack.png
Military science pack.png
Utility science pack.png
Space science pack.png
7 1,000 × 2^(N - F - 1) 1,000 2,000 4,000 8,000 16,000 ×2
Artillery shell range (research).png
Artillery shell range
+30% Artillery Shell Range
Automation science pack.png
Logistic science pack.png
Chemical science pack.png
Military science pack.png
Utility science pack.png
Space science pack.png
1 1,000 × 2^(N - F) 2,000 4,000 8,000 16,000 32,000 ×2
Artillery shell shooting speed (research).png
Artillery shell speed
+100% Artillery Speed
Automation science pack.png
Logistic science pack.png
Chemical science pack.png
Military science pack.png
Utility science pack.png
Space science pack.png
1 1,000 + 1,000 × 3^(N - F - 1) 2,000 4,000 10,000 28,000 82,000 ×3 then - 2000
Follower robot count (research).png
Follower robot count
+10 Follower Robots
Automation science pack.png
Logistic science pack.png
Chemical science pack.png
Military science pack.png
Production science pack.png
Utility science pack.png
Space science pack.png
7 100 × (N - F) + 900 1,000 1,100 1,200 1,300 1,400 +100

Cumulative cost

As the price of most infinite technologies (specifically, those based on geometric progressions) increases very steeply, it may be a good idea for players to set realistic target levels for each of the infinite technologies they wish to pursue, and make their factory plans accordingly. To that end, the following properties of cumulative infinite research prices may be useful:

  1. For infinite technologies whose underlying equation is a powers-of-two geometric series, the cumulative price of the first N - F infinite levels (skipping the first F level, so counting "infinite" levels only) is 2 × P[N] - P[F+1]; i.e., twice the price of the final researched level, less the price of the first "infinite" level.
    • As N increases, this is approximated well by 2 × P[N] = P[N+1], so the cumulative cost of researching to level N is about as much as researching level N+1.
    • If one decides a level M which one considers the "highest feasible" with their current science pack production capacity, expanding said capacity by a factor of X will allow about log[2](X) additional levels to be researched before the next level takes longer to research with the expanded capacity than level M + 1 would have taken with the pre-expansion production capacity.
    • For example, if one expands production capacity by a factor of 10, they will be able to research at least floor(log[2](10)) = 3 and at most ceiling(log[2](10)) = 4 additional levels in a given technology before the exponential increase in price negates the speed benefits of their ×10 capacity expansion.
  2. The cumulative price of the first N - F levels of infinite technologies whose underlying equation is an arithmetic series is (N - F) × (P[N] + P[F + 1]) ÷ 2; i.e, N - F times the mean of the prices of the first and last "infinite" level. For the Follower robot count (research), an additional 900 × (N-F) need to be added.
    • Expanding production capacity by a factor of X, as above, will in this case allow an additional N × (X - 1) levels to be researched before the benefit of the expansion is wiped out (i.e., research progress speed drops to or below what it was pre-expansion).
  3. The cumulative price of the first N levels of artillery shell shooting speed, the sole infinite technology whose underlying equation is a powers-of-three geometric series (equation type (2)) is 1.5 × P[N] - 0.5 × P[1]; i.e., 1.5 times the price of the final researched level, less half the price of the first level.

Note that these prices reflect research units, which will not be equal to science packs if productivity modules are used in labs. (In that case, the science pack requirement will be lower.)

See also